A disciplinarian is someone who expects you to follow a very strict set of rules. Many disciplinarians also believe in using harsh punishments for those who break those rules.
The word disciplinarian is all about discipline, which originally meant "punishment for the sake of correction." The Latin root, disciplina, was a lot more about teaching, instruction, and knowledge than punishment, though. Today, a disciplinarian's intent is both to instruct by means of explicit rules and to punish when those rules aren't followed. If your disciplinarian parents tell you to take the trash out, you're probably quick to follow their instructions.