Is a muqallid permitted to obey his father in matters that contradict his following of the more cautious opinion, or is he obligated to adhere to his taqleed regarding the necessity of acting upon the more cautious opinion?

Seeking the more cautious approach is commendable, and scholars refer to this as "exiting disagreement" (al-khurūj min al-khilāf). If by "more cautious" you mean exiting disagreement, then obedience to parents takes precedence because their obedience is obligatory, while the more cautious approach is commendable. However, if by "more cautious" you mean adhering to what the evidence proves to be obligatory, then it is not permissible to obey parents in abandoning it. A child is not required to switch from one madhhab to another if their father requests it. Disobeying parents is very serious, and one should not undertake it except after clarifying the permissible reason for it with scholars.