Can the state of California make a father give up his parental rights so he won't go to prison?

From that description, it is hard to see what is going on (or had been) or why.  While it does not sound precisely correct, that may have been his perception of perhaps interconnected legal events if, for example, there was a child abuse claim, a criminal prosecution, and a termination of parental rights case. If you know where everyone in question was located at the relevant time, a family law lawyer in that area might be able to do a search of court records (possibly online) to see if such a terminaiton was filed.