You should get a baptismal certificate, possibly stamped by the bishop. Chances are priests at other parishes aren't going to demand it but it's good to keep around. What if the convert is already baptized what do they get?
I was chrismated in an Antiochian parish and about a month later I received a signed, stamped certificate from Antiochian HQ in NJ - NOT from the parish.
Don't know how the OCA does it. Depending on the priest, if you move parishes (especially if out of state), there's a chance the priest might want you to prove you're Orthodox. I treat my chrismation certificate like I do my birth certificate and my social security card.
Interestingly, it is easier for me to get a new copy of my social security card, birth certificate, passport, license to kill at will, than my baptismal certificate from the backwater place I was baptized.
They have it. They won't just give it. lulz.
Or so it seems. Maybe the Pony Express is still in use or something out there.
When it couldn't get more strange, it has.
There has been more effort to certify my baptism than Obama's birth.
I have decided to take it as a sign that God doesn't want me in the Church.
I always assumed that to be the case, now I am getting evidence.