I once heard this: "Peter was only the head Apostle for the Jews, and Paul was the Apostle for the Gentiles." Noting that most Christians in the US would be considered “Gentiles”, the person summarized that, therefore, “Paul’s writings are the most important ones for us, and Peter’s writings are not required”. There was also mention that Peter is shown more in relation to preaching to the Jews, and Paul preaching to the Gentiles. But even if that were true, it’s not a logical step to say that, because “person A” preached mostly to “group Y”, he was therefore ONLY the Apostle for that group. That’s not only illogical, it’s also unbiblical.
According to another person, there is even a Christian group that claims we don't need the rest of the Bible at all, only what Paul wrote, because Paul is “our Apostle” and because, while Paul was converting Gentiles, the Jews rebelled against him, and so he stopped proclaiming to them.
Paul did indeed stop preaching to *some* Jews. In Acts 13: 13-43, Paul addresses the synagogue in Pisidia. (It's a beautiful address, and I recommend you take a moment to read it.) However, on the following sabbath, the Jews became jealous at the crowds who had gathered, and they contradicted Paul. Both Paul and Barnabas reprimand, "It was necessary that the word of God be spoken to you first, but since you reject it and condemn yourselves as unworthy of eternal life,we now turn to the Gentiles" (Acts 13:46).
But does this mean Paul stopped preaching to ALL Jews? No. Paul continued to preach to Jews wherever he went. Scripture tells us that this was his "usual custom" (Acts 17:2).
"When they took the road through Amphipolis and Apollonia, they reached Thessalonica, where there was a synagogue of the Jews. Following his usual custom, Paul joined them, and for three sabbaths he entered into discussions with them from the scriptures, expounding and demonstrating that the Messiah had to suffer and rise from the dead, and that “This is the Messiah, Jesus, whom I proclaim to you.” Some of them were convinced and joined Paul and Silas; so, too, a great number of Greeks who were worshipers, and not a few of the prominent women" (Acts 17:1-4). Of coarse, the Jews seemed to continually mess it up and would end up working against Paul. But this did not prevent him from preaching to the Jews.
God, Himself set Paul and Barnabas apart “for the work to which I have called them” (Acts 13:2). What work was involved in that? Preaching to the Jews in their synagogue at Salamis! “While they were worshiping the Lord and fasting, the Holy Spirit said, “Set apart for me Barnabas and Saul for the work to which I have called them.” Then after fasting and praying they laid their hands on them and sent them off. So, being sent out by the Holy Spirit, they went down to Seleucia; and from there they sailed to Cyprus. When they arrived at Salamis, they proclaimed the word of God in the synagogues of the Jews. And they had John also to assist them" (Acts 13:2-5).
Acts 14:1 places Paul and Barnabas in the Jewsish synagogue yet again, converting both Jews and Greeks: “The same thing occurred in Iconium, where Paul and Barnabas went into the Jewish synagogue and spoke in such a way that a great number of both Jews and Greeks became believers.”
This makes sense, of course, because God Himself said that Paul would be "an instrument whom I have chosen to bring my name before Gentiles and kings and before the people of Israel" (Acts 9:15). Paul would continue to preach to Jews up to, and including, the time of his imprisonment at Rome (see Acts 28).
Another passage flatly refutes the claim that Peter was “only the head Apostle for the Jews”. Acts 15:7 quotes Peter, with Paul and Barnabas present: "...Peter stood up and said to them, 'My Brothers, you know that in the early days of the Church, God made a choice among you, that I should be the one through whom the Gentiles would hear the message of the good news and become believers". In fact, the first Gentile converts were accepted and Baptized into the Church by Peter (Acts 10).
Peter's Episitles, part of Sacred Scripture, even refute such a claim. In Peter's first letter he specifically addresses his intended audience. "Peter, an apostle of Jesus Christ, to the chosen sojourners of the dispersion in Pontus, Galatia, Cappadocia, Asia, and Bithynia..." (1Peter 1:1). Some of the people to whom he is writing are the same people to whom Paul preached and wrote.
Why are Peter and Paul addressing the same people? Perhaps the best, and most common-sense, explanation can be found in the Scriptures themselves. We, as Christians, are ONE body in Christ (Rom 12; Eph 1,4; Col 1; Jn 10, 17 and a host of others), neither Gentile not Jew, slave nor free (1Cor 12). When we accept Christ, we are not defined by which race we came from as unbelievers, we are simply Christians, and the offspring of Abraham (Gal 3:27-29).
Paul says it well in 1Cor 15:1-11.
"I am reminding you, brothers and sisters,
of the Gospel I preached to you,
which you indeed received and in which you also stand.
Through it you are also being saved,
if you hold fast to the word I preached to you,
unless you believed in vain.
For I handed on to you as of first importance what I also received:
that Christ died for our sins in accordance with the Scriptures;
that he was buried;
that he was raised on the third day in accordance with the Scriptures;
that he appeared to Cephas, then to the Twelve.
After that, he appeared to more than five hundred brothers at once,
most of whom are still living, though some have fallen asleep.
After that he appeared to James,
then to all the Apostles.
Last of all, as to one born abnormally,
he appeared to me.
For I am the least of the Apostles,
not fit to be called an Apostle,
because I persecuted the Church of God.
But by the grace of God I am what I am,
and his grace to me has not been ineffective.
Indeed, I have toiled harder than all of them;
not I, however, but the grace of God that is with me.
Therefore, whether it be I or they,
so we preach and so you believed."