I cited earlier a Pope who recognised Antioch as a See of Peter. I apologise if you missed it.
Peter founded many Sees
In its present (ninth-century) form the "Martyrologium Hieronymianum" gives a second feast of the Chair of St. Peter for 22 February, but all the manuscripts assign it to Antioch, not to Rome
CATHOLIC ENCYCLOPEDIA: Chair of Peter
"the Chair of Saint Peter at Antioch, Saint Gregory the Great (on whom De La Salle wrote a meditation for March 12) says that Saint Peter was regarded as the head of the Church in Antioch for seven years prior to his going to Rome. The feast is now called simply the Chair of Peter. It was celebrated as early as the fourth century in Rome itself."
http://www.cbhs.org/lasallian_resources/February_22.htm
Pope Gregory the Great recognized this
"Your most sweet Holiness has spoken much in your letter to me about the chair of Saint Peter, Prince of the apostles, saying that he himself now sits on it in the persons of his successors
...
Wherefore though there are many apostles, yet with regard to the principality itself the See of the Prince of the apostles alone has grown strong in authority, which in three places is the See of one
...
He himself stablished the See in which, though he was to leave it, he sat for seven years. Since then it is the See of one, and one See, over which by Divine authority three bishops now preside, whatever good I hear of you, this I impute to myself. "
To Eulogius, Bishop of Alexandria
Book VII, Epistle XL
NPNF-212. Leo the Great, Gregory the Great | Christian Classics Ethereal Library
That is, they both sit in the Chair of Peter. In fact there's three chairs.