Paul was the Apostle of the Gentiles and he was the firsr believer to preach the "gospel of grace" to the Gentiles. The gospel which Peter preached Cornelius was the same gospel which was preached to the Jews (Acts 10:36-37) and the same gospel which Paul preached to the Jews and the promise of that gospel was foretold by the OT prophets:
"Paul, a servant of Jesus Christ, called to be an apostle, separated unto the gospel of God, (Which he had promised afore by his prophets in the holy scriptures,) Concerning his Son Jesus Christ our Lord, which was made of the seed of David according to the flesh; And declared to be the Son of God with power, according to the spirit of holiness, by the resurrection from the dead" (Ro.1:1-4).
Again, this gospel was promised by the OT prophets and the heart and soul of this gospel is the truth that Jesus is the Christ, the Son of God. The Lord Jesus fulfilled the "gospel" (good news) concerning the coming Messiah of which the prophecies spoke.
On the other hand the gospel which Paul preached to the Gentiles was a "mystery" truth, or something which had been kept secret since the world began.
"Now to him that is of power to stablish you according to my gospel, and the preaching of Jesus Christ, according to the revelation of the mystery, which was kept secret since the world began" (Ro.16:25).
You have probably been taught that only one gospel was preached during the Acts period and that is probably the source of your confusion. Common sense dictates that a gospel which was promised in the OT cannot possibly be the same gospel which was kept secret since the world began. And when you realize that the gospel which Paul preached to the Gentiles was different from the one Peter preached to the Gentiles and that the one which was preached to the Gentiles was the "gospel of grace" then it becomes obvious by Paul's own words concerning the "dispensation" which was given to him that the present dispensation did not begin until he began to preach that gospel to the Gentiles:
"If ye have heard of the dispensation of the grace of God which is given me toward you" (Eph. 3:2).
"Whereof I am made a minister, according to the dispensation of God which is given to me for you, to fulfil the word of God" (Col.1:25).
"...a dispensation of the gospel is committed unto me" (1 Cor.9:17).
The "dispensation" which was committed to Paul is in regard to "God's grace", a "ministry", and a "gospel." Here Paul sums up his dispensational responsibility:
"But none of these things move me, neither count I my life dear unto myself, so that I might finish my course with joy, and the ministry, which I have received of the Lord Jesus, to testify the gospel of the grace of God" (Acts 20: 24).
The Acts record will be searched in vain for any evidence that Peter preached the gospel of grace on the day of Pentecost. Instead, the gospel he preached on that day was the same one foretold by the OT prophets. There is ZERO evidence that the present dispensation began on the day of Pentecost.