Is it permissible to share in the sacrifice of one's father, which is designated as a charity, with the intention of it being an Uḍḥiyah (sacrificial offering during Eid al-Adha)?

Sharing one sheep as an udhiyah (definition) is not permissible, for a sheep is sufficient for only one person and their household. However, if you share living expenses, then one sheep is sufficient for all of you. It is permissible to offer a sacrifice on behalf of the deceased, whether they made a bequest for it or not, and similarly, to include them in the reward of the udhiyah offered by a living person. It is permissible for a father to be given money to buy a sacrifice and offer it on behalf of himself and his household, and he may include whomever he wishes in its reward.