Euthymios has supplied quite a list of the Divine Liturgies which honor Peter as the head of the choir, prince of the apostles, bishop of bishops etc.
Euthymios provided texts selectively to back his view, and ignored many others. As you have not supplied any liturgical material to back your case, here is a large proportion of the text for the Vigil of Apostles Peter and Paul. I apologise for the length of this post, however I felt that it would be more useful to quote extensively, lest I be accused of "quoting proof-texts", or "quoting out of context". The text is quite clear in ascribing equal
regard and honour to Apostles Peter and Paul. There is a distinct and beautiful complementarity between the verses for each apostle:
At “Lord, I have cried”:With what beauties of hymnody should we sing the praise of Peter and Paul? The wings of the knowledge of God, who flew through the ends of the earth and were lifted up high to heaven; the hands of the Gospel of grace, the feet of the truth of the proclamation, the rivers of wisdom, the arms of the Cross, through which He has cast down the arrogance of demons, Christ our God, who has great mercy.
With what spiritual songs should we praise Peter and Paul? The sharp mouths of the dread sword of the Spirit that slaughter godlessness; the radiant ornaments of Rome; the delights of the whole inhabited world; the reasoning tablets, written by God, of the New Testament, which in Zion Christ proclaimed, who has great mercy.
At the Litia:Come then today with fervour, acceptable sacrifice of the faithful; as we stand together in choir, let us garland with fitting songs Peter and Paul, the chosen weavers of grace; because they sowed the word unstintingly for all and enriched them with the gift of the Spirit; and being branches of the true vine, they have brought to perfection for us the ripe grape cluster, making our hearts glad. To them let us cry out, with faces unveiled and with pure consciences, as we say: Hail, guides of the unreasoning and servants of those with reason. Hail, fair chosen members of the Maker and Guardian of all. Hail, protectors of the good and persecutors of the deceitful. Let us implore them to ever entreat the Creator and Teacher to give the world stable peace and our souls His great mercy.
Let us, the whole world, praise as its champions the Disciples of Christ and foundations of the Church, the true pillars and bases, and inspired heralds of the doctrines and sufferings of Christ, the Princes, Peter and Paul. For they passed through the whole breadth of the earth as with a plough, and sowed the faith, and they made the knowledge of God well up for all, showing forth the understanding of the Trinity. O Peter, rock and foundation, and Paul, vessel of choice; the yoked oxen of Christ drew nations, cities and islands to knowledge of God. While they have brought Hebrews again to Christ and intercede that our souls may be saved.
Peter, Prince of the glorious Apostles, the rock of the faith, and inspired Paul, the preacher and beacon of the holy Churches, as you stand before the throne of God, intercede with Christ on our behalf.
Paul, mouth of the Lord, foundation of doctrines, once the persecutor of Jesus my Saviour, but now become first-throned of the Apostles, blessed one; therefore you saw things ineffable, O wise one, when you ascended to the third heaven, and you cried: Come with me, and let us not be deprived of the blessings.
The citizens of the Jerusalem on high, the rock of the faith, the preacher of the Church of Christ, the pair of the Trinity, the fishers of the world, leaving behind today the things on earth, have journeyed in truth to God, and they implore Him with boldness that our souls may be saved.
As the wisdom of God, the co-eternal Word of the Father, foretold in the Gospel, you are the fruitful branches, all-praised Apostles; you bear on your shoots the ripe and lovely grape cluster, which we faithful eat and experience a taste which brings delight. Peter, rock of the faith, and Paul, boast of the inhabited world, establish the flock which you have gained by your teachings.
At the Apostikha:Who will recount your chains city by city and your afflictions, glorious Apostle Paul? The toils, the pains, the vigils, the sufferings from hunger and thirst, from cold and nakedness, the basket, the beatings, the stonings, the journeying, the deep, the shipwrecks? You became a spectacle to Angels and to humans. You endured all things in Christ who gave you power, that you might gain the world for Christ Jesus, your Lord. And so we beseech you, as we faithfully celebrate your memory, intercede without ceasing that our souls may be saved.
Verse: Their sound has gone out into all the earth; and their words to the ends of the world.
Who will recount your chains city by city and your afflictions, glorious Apostle Paul? Or who will set down the struggles and the toils, by which you toiled in the Gospel of Christ, that you might gain all people and bring the Church to Christ? But ask that she may keep safe your fair confession until her last breath, O Paul, Apostle and teacher of the Churches.
Verse: The heavens declare the glory of God; and the firmament proclaims the work of his hands.
Let us praise Peter and Paul, the great beacons of the Church; for they shone more brightly than the sun in the firmament of the faith, and with the rays of the proclamation they brought the nations from ignorance to the knowledge of God. The one was nailed to a cross and so found the path to heaven, where he was handed the keys of the kingdom by Christ; while the other was beheaded by the sword and so departed to the Saviour and is fittingly called blessed. Both denounced Israel for stretching out his hands unjustly against the Lord Himself. And so at their prayers, Christ our God, cast down those who are against us, and strengthen the Orthodox faith, as You love mankind.
A joyous feast has shone out today on the ends of the earth, the all-honoured memorial of the wisest Apostles and their princes, Peter and Paul; and so Rome dances and rejoices. Let us also, brethren, celebrate in songs and psalms this all-revered day, as we cry out to them: Hail, Peter, Apostle and true friend of your teacher, Christ our God. Hail, Paul, well-loved, herald of the faith and teacher of the inhabited world. Holy pair, chosen by God, as you have boldness, implore Christ God that our souls may be saved.
Troparion of the Feast:First-throned of the Apostles and teachers of the inhabited world, intercede with the Master of all things to give peace to the world and to our souls His great mercy.
Sessional hymn:Let us, the faithful, fittingly praise the champion Paul, the net of the world, and the most-praised Peter, the rock of the Church, who holds the keys of heaven, for the universe has been enlightened by them with the faith of the Trinity. Glory to Him who has glorified you, glory to Him who has strengthened you, glory to Him who because of you has given us everlasting life.
Magnification:We magnify you, O Peter and Paul, Apostles of Christ, who enlightened the universe with your teachings, and led the ends of the earth to Christ.
Sessional hymn after the Polyeleos:Let us praise Peter and the all-wise Paul, the great and radiant luminaries, who were shown to be pre-eminent among the disciples; for shining forth with the fire of the divine Spirit, they burned away all the gloom of deception. Therefore they have fittingly shown to be warriors of the Kingdom above, equally enthroned in grace. For this cause we exclaim: O Apostles of Christ our God, ask forgiveness of sins to those who with love celebrate your holy memory.
Sessional hymn after psalm 50:Let us honour with hymns of praise the true preachers of piety, the all-radiant stars of the Church; Peter, the rock of faith, and Paul, the teacher of the truth and initiate of the mysteries of Christ. Having sown the word of truth in the hearts of the faithful, may they both entreat Christ our God, who gives abundantly to all, that our souls be saved.
There are two canons sung at this feast, one for each apostle. Some examples of complementary troparia:
From Ode 1:Having foreknown you, O most-blessed Peter, the Pre-eternal One ordained you of old as the leader of His Church, the first-enthroned.
When Christ called existence out of non-existence, O most-blessed Paul, He chose you from your mother’s womb to carry His divine name which is above every name before the nations, for He has been gloriously glorified.
From Ode 3:On the rock of your theology, Jesus the Master established the unassailable Church, and there we glorify you, O Apostle Peter.
You have been set as a precious foundation stone for the souls of the faithful, a cornerstone of the Saviour and Lord.
Ypakoi:What dungeon did not hold you prisoner? What Church did not have you as an orator? Damascus extols you, O Paul, for it knew you blinded by the Light; and Rome, which received your blood, boasts in you; but Tarsus, your birthplace, rejoices yet more with love and honour. O Peter, rock of the Faith, and Paul, boast of the whole world, coming together from Rome, make us steadfast.
Kontakion:The steadfast and divinely eloquent preachers, the foremost of Your apostles, O Lord, You have received into the delight of Your good things and into rest; for You have accepted their sufferings and death as greater than any whole-burnt offering, You who alone knows the hearts of men.
From Ode 7:Having risen from the dead, Christ appeared to you, the first-called and most beloved, as the right glorious one who was first to preside over the apostles. ***
He who in latter times was called and surpassed all others in zeal, became the seal and crown of Your apostles, O Christ. With him the people of the Church sing to You: O God of our fathers, blessed are You.
(*** A note on “first to preside over the apostles”: Peter was the first to testify at the first Apostolic council at Jerusalem, which was presided over by James, Brother of the Lord, and Bishop of Jerusalem. As scripture and history shows, he may have been given primacy of honour among his fellow apostles, but this did not mean that his word would automatically prevail on the matters of faith being discussed. For instance, his position on the matter of circumcision prior to baptism did not prevail, following Paul’s eloquent and inspired repudiation of it.)
From Ode 8:O Christ who announced to Peter that the nations were cleansed, by spiritual radiance purify my thoughts, for I cry: Bless the Lord, all you works of the Lord.
You betrothed the Church to Christ, presenting it to Him as a bride; for you were shown to be its wedding-escort, O God-bearing Paul. Therefore, it glorifies you for all ages.
Exaposteilarion:Let us all hymn the foremost among the apostles, the godly Peter and Paul, the universal luminaries, the preachers of the Faith, the divinely sounded clarions, the speakers of dogmas, the pillars of the Church and destroyers of falsehood.
From the Praises:The most honoured feast of the apostles has arrived for the Church of Christ, bringing salvation to us all. Therefore, mystically weaving hymns for them, let us say: Hail, O lamps to those who are in darkness, shedding forth rays of the spiritual Sun! Hail, O Peter and Paul, unshakeable foundations of the divine doctrines, friends of Christ, precious vessels! Come into our midst, bestowing immaterial gifts to those who praise your feast with hymns.
There are two simple and very accessible sources of doctrine which are consistent across the whole Orthodox world: iconography, and liturgical texts. The Orthodox Church regards Apostles Peter and
Paul as the pre-eminent apostles, not just Peter. This is abundantly clear in both iconography, and in the text for the vigil of Sts Peter and Paul. Both are held in equal regard, hence the joint feast-day. The joint commemoration has a purpose: to remove
any notions of supremacy. Another example of this is the feast of the Three Hierarchs, instituted to show the Church's equal regard for Sts Basil the Great, John Chrysostom and Gregory the Theologian.
The feast for Sts Peter and Paul dates back to no later than the mid-8thC, given that the canons at Matins were written by St John of Damascus, and other hymns by his contemporaries, including St Andrew of Crete.
Individual fathers may contradict each other, but the liturgical texts represent the distillation, the essence, the core, of the teachings of the Church for any feast. It is what the whole
Orthodox Church espouses and proclaims, irrespective of geographic location or jurisdiction. It is the same with iconography. Individual icons of Apostle Peter exist in great numbers, but there is no indication at all that he is supreme over all the others. In icons of the assembly of the Twelve Apostles, Peter and Paul are in the foreground, often jointly holding a model of a church building, a motif also used in icons of saints regarded as "equals to the Apostles", such as the enlightener-saints e.g. Sts Vladimir and Olga of Kiev, Constantine and Helen, Nina of Georgia, Olaf of Norway, etc. Apostle Andrew is shown standing behind Peter and Paul, but still in a position of prominence, as he was the first to be chosen by Christ to be a disciple. All of this is completely consistent with Orthodox teaching.
Every single Orthodox church in the world uses the same text as this when celebrating this feast. Only the liturgical language used will vary. In the light of the liturgical and iconographic tradition, is it still possible for an Orthodox Christian to honestly and unreservedly conclude that Peter has the higher authority over all the other apostles, including Paul?