Is it obligatory to obey parents in every permissible matter, even if it entails psychological or physical hardship, or if it prevents a permissible matter that causes no harm to the parents?

Obeying one's parents is obligatory in all that they command, as long as it does not involve disobedience [to God] or harm. You must obey your father regarding the study time he has set and refrain from entering the websites he has forbidden you from. As for the psychological harm you mentioned, if it is true and not an exaggeration, then strive to explain it to your father so he may lighten your burden.