Is it incumbent upon one who intends to slaughter an Udhiyah on behalf of his father to do something, or upon his father to do something?

It is Sunnah for your parents to refrain from taking hair and nails if the crescent of Dhul-Hijjah is sighted until they slaughter their sacrifice. This is based on the saying of the Prophet, peace and blessings be upon him: "When you see the crescent of Dhul-Hijjah and one of you intends to offer a sacrifice, he should refrain from (cutting) his hair and nails." The wisdom behind this is for the entire body to remain intact so that all its parts may be freed from the Fire. The dislike (of cutting) is removed if there is a need. As for you, you are not required to refrain from this unless you intend to include your parents in the sacrifice. In that case, you should refrain from taking from your nails and hair, but those whom you include with you in the reward of the sacrifice from your family are not required to refrain from that.