As recorded in Eusebius Church History 7.7-9:
His [Dionysius of Alexander] fourth epistle on baptism was written to Dionysius of Rome, who was then a presbyter, but not long after received the episcopate of that church. It is evident from what is stated of him by Dionysius of Alexandria, that he also was a learned and admirable man.Now I won't make too much of this account. While the bishop of a Christian center like Alexandria writing continuously not to one but to two successive bishops of Rome isn't to be lightly shrugged off, it would not have been unusual at all for two bishops of perfectly equal status to ask one another for advice. In the second passage quoted above, Dionysius refers to him as "brother" and I bring this up to be fair in regards to my earlier posts regarding the use of the word "father" in reference to the bishop of Rome during the early centuries of the Church here and here. As I said then:
His fifth epistle was written to Xystus, bishop of Rome. In this, after saying much against the heretics, he relates a certain occurrence of his time as follows: For truly, brother, I am in need of counsel, and I ask your judgment concerning a certain matter which has come to me, fearing that I may be in error.
Besides these there is also extant another epistle of the same man on baptism, addressed by him and his parish to Xystus and the church at Rome. In this he considers the question then agitated with extended argument. And there is extant yet another after these, addressed to Dionysius of Rome, concerning Lucian.
"Calling another bishop "brother" would certainly not amount to Gaul denying the 'honorary primacy' of Rome (if one were to argue such a point)"And here is one example of such a thing happening. I have yet to run across any bishop referring to any other bishop (other than the bishop of Rome) as "father" yet. I would be interested to learn if there were any instances of that.