I don’t think Peter speaks of his primacy directly in the epistles. But he does speak of obedience often, including in the second verse of 1 Peter.
***Peter, an apostle of Jesus Christ, to the chosen sojourners of the dispersion in Pontus, Galatia, Cappadocia, Asia, and Bithynia, in the foreknowledge of God the Father, through sanctification by the Spirit, for obedience ***and sprinkling with the blood of Jesus Christ: may grace and peace be yours in abundance (1 Peter 1:1-2)
***Since you have purified yourselves by obedience to the truth ***for sincere mutual love, love one another intensely from a [pure] heart. (1 Peter 1:22)
I like that one… obedience to the truth.

I’d like to hear an Evangelical pastor preach on that verse.
Peter speaks of the rock or stone in the second chapter and exhorts his readers to obey institutions and civil authorities which possess legitimate authority. It is easy to see how he also infers that his readers should obey the legitimate authority of the Church but I might be reaching a bit there. It is interesting however, to read the beginning of 1 Peter 5 which goes into the structure of authority in Church hierarcy and how prespyters and younger men are to conduct themselves in light of that hierarchy.
And Peter’s second letter, which some atribute to a proteje as late as the year 120 has this amazing passage which all Catholics are familiar with:
Moreover, we possess the prophetic message that is altogether reliable. You will do well to be attentive to it, as to a lamp shining in a dark place, until day dawns and the morning star rises in your hearts. Know this first of all, that there is no prophecy of scripture that is a matter of personal interpretation, for no prophecy ever came through human will; but rather human beings moved by the holy Spirit spoke under the influence of God. (2 Peter 1:19-21)
But I just read all of 1 and 2 Peter and didn’t see anything overt.
-Tim-