The Bishop of Rome was considered the preeminent by all the other major sees. Its just whether he was first among equals or outright leader that was the controversy. So the RCC has a bit of a claim as the parent..
That's the only reason the Roman diocese became dominant. Not any sort of lineage or pedigree.
There's a very informative youtube video about the schism: https://www.youtube.com/watch?v=a6rWf0k8d78