All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
It is not permissible to pay Zakaah to your father to perform Hajj or for anything else, whether he is poor or rich. If your father is rich, he is not entitled to receive Zakaah; if he is poor, it is your duty to suffice his need.
Moreover, if you give Zakaah 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 Zakaah to yourself; it is not permissible. Know that, according to the majority of Muslim scholars, it is not permissible to pay Zakaah to a poor person to perform Hajj. Only Imam Ahmad, in one of his narrations, legalizes such paying of Zakaah. 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 Zakaah to your father.
Allah knows best.