Praise be to Allah.
If the father took on this debt in order to spend on himself, it is not permissible for his son to give his zakaah to his father to pay off his debt, because the son is obliged to spend on his father.
But if this debt was for something else other than spending on himself, then there is nothing wrong with the son paying it off with zakaah, because he is not obliged to pay off his father’s debt, so he is not avoiding an obligatory duty by doing so.
Shaykh Ibn Baaz said:
Paying zakaah to relatives who are entitled to it is better than giving it to those who are not your relatives, because charity given to relatives is both charity and upholding the ties of kinship, unless these relatives are those on whom you are obliged to spend, in which case giving zakaah to them in order to free yourself from the obligation to spend on them is not permissible. If it so happens that these brothers and sisters whom you have mentioned are poor, and you cannot afford to spend on them, then there is nothing wrong with you giving them your zakaah. Similarly, if these brothers and sisters owe debts to people and you pay off their debts with your zakaah, there is nothing wrong with you doing that. That is because a person is not obliged to pay off the debts of his relative, so if he pays them off with his zakaah, that is something for which he will be rewarded. Even if it is your son or father who owes a debt to someone and cannot pay it off, it is permissible for you to pay off your father’s debt from your zakaah, i.e., it is permissible for you to pay off your father’s debt from your zakaah and it is permissible to pay off the debt of your son from your zakaah, so long as the reason for this debt is not to obtain money for maintenance that it is your duty to pay. If the reason for it is to obtain maintenance which it is your duty to pay, then it is not permissible for you to pay off the debt from your zakaah, lest you use that as a trick to get out of spending on those on whom you are obliged to spend so that they can take on debts and you can pay them off with your zakaah. End quote.
Majmoo’ Fataawa Ibn Baaz, 14/310
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said:
If you are obliged to spend on someone, it is not permissible for you to give zakaah to him as maintenance, but if it is to pay off his debt, there is nothing wrong with that. If we assume that a father owes a debt, and his son wants to pay off his debt from his zakaah, and (the father) is unable to pay it off, there is nothing wrong with that. The same applies to a mother or a son. But if you are giving zakaah for the purpose of maintenance, that is not permissible, because by doing this you are saving your money, but it is obligatory to spend on one's parents, mother and father, and on sons and daughters, and on everyone from whom you would inherit if he died, i.e., everyone from whom you would inherit if he died, you are obliged to spend on, because Allaah says (interpretation of the meaning):
“And on the (father’s) heir is incumbent the like of that (which was incumbent on the father)”
Allaah has enjoined upon the heir the payment for breastfeeding, because breastfeeding is like spending. End quote.
Fataawa Ibn ‘Uthaymeen, 18/416.