It is permissible to give zakaah to one who wants to get married, if he cannot afford to get married. We have explained that in the answer to question no. 21975.
It is also permissible to give zakaah to one who is in debt, if he cannot afford to pay off his debt.
It is not permissible for a father to give zakaah to his son, because if the son is poor the father is obliged to spend on him. Similarly, if he needs to get married, the father is obliged to arrange his marriage. Giving zakaah to him is evading this obligation, so it is not permissible. But the father may give his zakaah to a son who is in debt, because he is not obliged to pay off his son’s debt, unless the son borrowed money because of the maintenance that his father should have provided for him in the first place.
Shaykh Ibn Baaz (may Allaah have mercy on him) said: It is permissible to pay off your father’s debt with your zakaah, and it is permissible to pay off your son’s debt with your zakaah, so long as the reason for this debt was not to get maintenance that you are obliged to pay. If the reason for it was to get maintenance that you are obliged to pay, then it is not permissible for you to pay off this debt with your zakaah, lest that be taken as a trick to avoid spending on those on whom you are obliged to spend by letting them take a loan then paying it off from the zakaah. End quote from Majmoo’ Fataawa Ibn Baaz (14/310).
But if the father’s wealth is not sufficient to arrange a marriage for his son, in that case it is permissible for him to give his zakaah to his son to help him with the expenses of marriage, because what the father cannot afford, such as the expenses for his son’s marriage, is not obligatory upon the father, because he is not able to afford it. In that case, giving zakaah to the son is not evading something that is obligatory for the father.
This has been explained in the answer to question no. 85088.
If it is permissible for the father to give his zakaah to pay off the son’s debt, then he should give him the money, after which the son should pay off the debt. It is not permissible for him to waive the debt and regard that as directly giving zakaah, because zakaah involves taking and giving, as Allaah says (interpretation of the meaning):
“Take Sadaqah (alms) from their wealth in order to purify them and sanctify them with it”
And the Prophet (peace and blessings of Allaah be upon him) said: “Allaah has enjoined upon them charity (zakaah) to be taken from their rich and given to their poor.” Waiving a debt for a poor person is neither taking nor giving, and what is owed by the poor person is something that is not present and is not tangible, therefore it cannot be regarded as being like wealth that is present and tangible. Because what is owed is of less importance in the mind of a person than what is present and tangible, so paying the debt on behalf of the poor person is like giving him something of lesser value.
End quote from Majaalis Shahr Ramadaan by Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him), al-Majlis 17.
Conclusion: Your father may give his zakaah to you, because you are in debt, so you may take it from him then pay off the debt, so long as you did not take the debt to make up for the maintenance that your father is obliged to provide.
And it is permissible for him to give zakaah to you if he cannot afford the expenses for you to get married.
And Allaah knows best.