How can a father be persuaded to approve his daughter's marriage to a young man the family rejects, despite the mother's approval and the couple's desire?

If the young man is religious and of good character, then marry him, for the Prophet, peace and blessings be upon him, said: "If someone whose religiosity and character you approve comes to you seeking marriage, then marry him. If you do not do so, there will be tribulation on earth and widespread corruption." Your father does not have the right to object to your marriage to him without a legitimate reason, and material or social differences do not justify rejection. Try to convince your father through supplication, or through the intercession of close relatives. If he is not convinced, you may refer the matter to a Sharia judge to marry you if the withholding (عضل) is proven. If you are patient and prioritize your father's رضا (pleasure), Allah may bless you with someone better.