Is it obligatory for a son who stole from his father to confess to him and return the money directly, or can he offer a gift or grant a donation, and what is required of him concerning stealing from his mother in case of doubt?

It is not obligatory to confess to the father about stealing his money, but the stolen item must be returned to him by any means. It is more prudent that the return not be in the form of a gift. As for doubting the theft of money from the mother, nothing is obligatory regarding it, because the default is the acquittal of the ذمة (financial liability/conscience).