Christianity never gained the same strength in Persia as it did within the Roman Empire, Zoroastrianism remained predominate until the Islamic conquest. There were Christians there, but they were culturally and politically distinct from the Romans, making ties tenuous at best, the political tension came to a head under theological pretenses at the Council of Ephesus when they formally broke communion from the Christians to the west.
To summarize a large breadth of history:
On the contrary. The Persian church---simply self-identifying as the Church of the East---was far from being small, and not altogether comprised of Persians. Frequent moments in Persian history witness a greater Christian population than the religion of their leaders. Zoroastrianism remained predominate in the sense of being the "state religion" prior to the dawn of Islam, but its numbers were frequently stifled by the outnumbering Christians, who were in fact Persian. Records do show the Church of the East going through great pains trying to convert Syriac communities, but the Syriac Orthodox vehemently "held their own" for all intents and purposes. The COE's numbers were at one time larger than any other Christian body on the planet, though formally and politically silenced by the Persian King and Emperor's frequent wars. Small amounts of communication between Seleucia-Ctesiphon and Constantinople and Rome existed up until the 8th century, more charitable than one would expect.
Granted, the COE was a subsidiary of the Patriarch of Antioch, though after Ephesus and the formal break in communion, the Catholicose of Seleucia-Ctesiphon eventually renamed himself a Patriarchate, similar to the Maronites' action in appointing their own Patriarch during Antioch's woes at the time. At one point, the the Patriarch of Seleucia-Ctesiphon had 23 metropolitan sees, and his reach went as far to Mongolia, China, and south to India, fertilizing their Christian community with the Malabarese "Chaldean" or "Assyrian" based tradition. Communication and in some documented cases authority with the Malabarese is recorded up until the arrival of the Portugese. In the 9th century, post-Islam, while still recovering from one of the largest single genocides of Christians on record, the Patriarchate was transfered to Babylon, earning them the name "Church of the Martyrs" for their immense loss of life. Their numbers have never recovered, and have all but become worse to the Iraqi Occupation and subsequent violence of the Iraqi insurgent movement.
The glory of the Church of the East was immense, formidable to the unified East and West Christians across the Tigris, however brief. Its remnants after Islam were fraught with hereditary patriarchs, Byzantinesque politics, and an unfortunate lack of leadership, and their future is as uncertain as it always has been.