What is the ruling regarding someone who swore multiple times not to wear a new garment for Tarawih prayers, and then went back on his oath?

Your father incurs no sin by going to the mosque in his regular clothes if they are suitable for him. He should not be embarrassed or forced into something he does not desire, as new clothes are not a requirement for every prayer. If he wishes to atone for his oath and comply with your wish, he may do so. You can pay the expiation on his behalf with his consent, or he can pay it himself. This is especially true if the clothing is not suitable for him, based on the Prophet Muhammad's (peace be upon him) saying: "Whoever takes an oath and then sees something better than it, let him do that which is better and atone for his oath."