Not sure this should be an "answer" or part of the OP, but I'll put it here for now.
To draw out a comment from the Apostolic Constitutions, Book Vii, P 46, that is mentioned in an answer to OP, it says this (bold mine):
Of Antioch, Euodius, ordained by me Peter; and Ignatius by Paul. Of Alexandria, Annianus was the first, ordained by Mark the evangelist; the second Avilius by Luke, who was also an evangelist. Of the church of Rome, Linus the son of Claudia was the first, ordained by Paul; 2 Timothy 4:21 and Clemens, after Linus' death, the second, ordained by me Peter. Of Ephesus, Timotheus, ordained by Paul; and John, by me John. Of Smyrna, Aristo the first; after whom Stratæas the son of Lois; 2 Timothy 1:5 and the third Aristo. Of Pergamus, Gaius. Of Philadelphia, Demetrius, by me. Of Cenchrea, Lucius, by Paul. Of Crete, Titus. Of Athens, Dionysius. Of Tripoli in Phœnicia, Marathones. Of Laodicea in Phrygia, Archippus. Of Colossæ, Philemon. Of Borea in Macedonia, Onesimus, once the servant of Philemon. Of the churches of Galatia, Crescens. Of the parishes of Asia, Aquila and Nicetas. Of the church of Æginæ, Crispus. These are the bishops who are entrusted by us with the parishes in the Lord; whose doctrine keep always in mind, and observe our words.
There's no sense or reason given as to why Rome would later claim it was the Seat of Peter, as opposed to Antioch, whose first bishop was ordained by Peter. For Rome, it was the second bishop. This implies that Antioch would be the head (if one agrees with this assumption in the first place).
As well, the last sentence from the quote basically confirms that all parishes are in the Lord. There's no sense of one above the others, be it Antioch or Rome.
So, OP still looking for an answer.