Is it permissible for someone to donate their Zakat to a wealthy father, and then for the father to use this money to perform Umrah?

It is not permissible to give Zakat to someone who is not eligible for it. It must be given to those whom Allah Almighty has mentioned as eligible. If the father is in need of Zakat, then giving it to him is sufficient, and he is allowed to use it to fulfill his needs, such as Hajj and Umrah. However, if the father is not in need, and the son gives it to him solely for Hajj or Umrah, then it is not sufficient.