Is it obligatory to leave mixed-gender work in the hospital despite the parents' refusal, for fear of losing one's faith and committing an act of disobedience?

If you and your parents are in need of the work, then continue with it, avoid reprehensible acts, and strive to minimize evil as much as possible. If you are not in need of it, then there is no sin upon you for leaving it, but convince your parents of that so they do not become angry. However, if your leaving the work angers them, then continue with it until they are convinced.