In Acts 22:28, Paul says, "I was free born," in reference to him being born a Roman citizen, but I was under the impression Roman citizenship had to be earned. One had to either buy it, or work for it through things like military service and such. Were there ways around this? Like maybe his parents bought it for him or something? Or am I just mistaken?
but I was under the impression Roman citizenship had to be earned.
This is incorrect. You were a Roman citizen (cive) if both of your parents were Roman citizens (cives). You could also be born a citizen if one of your parents was a non-citizen (peregrinus) but had a recognized marriage contract with the other parent. That is, if your citizen father begot you on a slave woman, you would not be born a citizen. "Buying" Roman citizenship is a later thing; the only sources of which I'm aware that mention it are Acts* and the 3rd century history by Cassius Dio who says that Claudius started selling citizenships after he became emperor.** That would have been too late for Paul's parents to have bought it for him, as he was born probably around AD 5 or 6, and anyway would contradict his statement that he was born a citizen. It's pretty certain, then, that one or both of his parents were citizens.
* 22:27-28, in the same discussion you mention in which Paul says he's a citizen. He is talking to a centurion who claims he (the centurion) bought his citizenship.
** Historia Romana 60.17