Is it permissible for a son to give his father money secretly, without his mother's knowledge, given that the father will return it?

The money your mother entrusted to you is a trust (amanah); it is not permissible to dispose of it except with her permission. If she permits giving a loan or a gift from it to your father, then there is no problem. Otherwise, it is not permissible. And the financial support of a poor father is obligatory upon his well-off children.