Obey, comply, mind are comparable when they mean to follow the wish, direction, or command of another.
Obey is the general term and implies ready or submissive yielding to the authority of another (as by the performance of his command or bidding) or subjection to a higher principle or to the agency, force, or impulse by which it is actuated.
Comply, often with with, usually carries a stronger suggestion than obey of giving in to or yielding to a person's desires or expressed requests, the rules of an organization, the requirements of a law, or the conditions of one's environment. It therefore often comes close to conform and accommodate and tends to imply complaisance, dependence, or lack of a strong opinion.
Mind, though often used in the sense of obey, especially in reference to children in a weaker sense carries the implication of heeding or attending to an expressed wish, demand, or command in order that one may comply with it.