Praise be to Allah, the Lord of the World; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.
It is not permissible to pay Zakah to your father to perform Hajj or for anything else, be he poor or rich. If your father is a rich, he is not entitled to receive Zakah; if he is poor, it is your duty to suffice his need.
Moreover, if you give Zakah to your needy father, it means that you will drop his rights on you. In other words, such an action is a form of paying Zakah to yourself; it is not permissible. Know that, according to the majority of Muslim scholars, it is not permissible to pay Zakah to a poor person to perform Hajj. Only Imam Ahmad , in some narration, legalizes such paying of Zakah. But, the opinion of the majority is the dominant one in this concern.
Anyway, the above-stated difference of opinion does not cover your father, i.e. you should not pay Zakah to your father.
Allah knows best.