Could a psychiatrist commit me to a mental hospital without my parents' permission?

Rarely, yes: If the psychiatrist is very concerned about your welfare and fears your parents will not get you the inpatient help you need, s/he may fill out commitment papers. In some unfortunate cases, parents might be drug-abusing or otherwise not able to be responsible -- and kids can harm themselves in such situations. An inpatient observation period followed by a hearing ; recommendations would result.