Is it permissible to gift the Udhiyah to a living, financially capable father, with the intention that the father would be rewarded for it if he slaughters it for himself and his household, knowing that the giver has not yet performed an Udhiyah for himself?

It is permissible for you to gift your father a sacrifice (udhiyah), whether he sacrifices on his own behalf or not, or whether he is wealthy or alive, or if you yourself have not sacrificed; none of these factors affect the permissibility of the gift or the validity of the sacrifice. Your father will receive the reward for the sacrifice if he performs it, as it has become his property, and you will receive the reward for showing kindness and gifting your father. You should sacrifice on your own behalf in the future, as the udhiyah is a confirmed Sunnah.