Is it permissible for a son to give his father from his Zakat money to pay off a debt owed to his mother, and is the mother entitled to dispose of this money as she wishes?

What the father gave to the mother, whether as a gift or personal allowance, becomes her property upon her taking possession of it. If he then borrows it from her, it is a debt that he is obligated to repay, unless she pardons him.

As for household expenses, the wife acts as an agent in managing them. Any surplus from these funds remains the property of the father, and it is not permissible to take it without his permission, as Allah Almighty says: (O you who have believed, do not betray Allah and the Messenger or betray your trusts while you know [the consequence].) And as the Prophet, peace be upon him, said: (The wealth of a Muslim person is not lawful [to take] except with his willing consent.)

One must be cautious about debt and not incur it unless there is an intention to repay it. For whoever takes people's money intending to repay it, Allah will help him repay it; and whoever takes it intending to destroy it, Allah will destroy him. Debt may prevent entry into Paradise until it is repaid.

If the father does not possess the means to repay his debt to your mother, he is considered an insolvent debtor (gharim), and it is permissible to pay his debt from zakat. This is an exception to the prohibition of giving zakat to ascendants and descendants.

It is permissible to repay the debt on behalf of an insolvent debtor without his knowledge, but it is preferable to hand over the money to the debtor if he is trustworthy so that he can make the payment himself. Otherwise, it should be paid directly to the creditor. It is not necessary to inform the father that the debt was paid from zakat. If you fear that he might borrow from it again, then do not inform him.