While Peter deserves the highest praise for his devotion to the Savior, it is quite clear in Scripture that the Rock is not Peter, but Christ.
Only five verses after Matthew 16:18 which you cite, Christ compares Peter to Satan for hindering his ministry: "But He turned and said to Peter, 'Get behind Me, Satan! You are an offense to Me, for you are not mindful of the things of God, but the things of men'." (Matthew 16:23)
Peter himself refers to Christ as the 'chief cornerstone' (1 Peter 1:4-6). Paul adds to this in Ephesians: " "Now therefore ye are no more strangers and foreigners, but fellowcitizens with the saints, and of the household of God; And are built upon the foundation of the apostles and prophets, Jesus Christ himself being the chief corner stone; In whom all the building fitly framed together groweth unto an holy temple in the Lord: In whom ye also are builded together for an habitation of God through the Spirit." (Ephesians 2:19-22) and again in 1 Corinthians 3:11: "For no other foundation can anyone lay than that which is laid, which is Jesus Christ."
Even David knew it: "Who is a rock, except our God?" (Psalm 18:31).
Bless your evening.
But; when it's been POUNDED into your head that it IS Peter...