Pharisees are not the same thing as priests. They were, in essence, a political/social/religious "movement" or "denomination" so to speak.
Only Levites (Jews of the tribe of Levi) could become priests. A Levite might be a Pharisee, and a priest might be a Pharisee, but not all Levites were Pharisees, and not all priests were Pharisees. Paul could not have been a priest at all, because he was not a Levite.