Praise be to Allah.
Offering the udhiyah is a confirmed Sunnah; for it to be prescribed and encouraged it is stipulated that the one who offers the udhiyah should be independent of means, in the sense that the price of the udhiyah is surplus to what he needs to suffice himself and those on whom he is obliged to spend.
Please see the answer to question no. 70291.
If a person does not own enough to pay the price of the udhiyah, surplus to what he needs for his expenses, then he is not obliged to offer an udhiyah.
If the members of a family share accommodation and expenses, one udhiyah is sufficient on behalf of all of them.
The scholars of the Standing Committee said:
If the family is large, but they live in one house, then one udhiyah is sufficient on behalf of all of them, but if they offer more than one, that is preferable.
End quote from Fataawa al-Lajnah ad-Daa’imah (11/408)
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:
The inhabitants of one house may share one udhiyah even if they are many. If they are brothers who eat together and live in one house, then they may share one udhiyah, even if they have many wives. The same applies in the case of a father with his sons, even if one of them is married: they may share one udhiyah.
End quote from Majmoo‘ Fataawa wa Rasaa’il al-‘Uthaymeen (25/43)
See also the answer to question no. 96741
Based on that, if you live in the same accommodation with your father, then one udhiyah is acceptable on behalf of the entire household, including your father and your mother.
The scholars of the Standing Committee were asked:
If my wife lives with my father in the same house, is it acceptable to offer one sacrifice on ‘Eid al-Adha for myself and my father, or not?
If the situation is as you describe, and that the father and his son live in one house, then one udhiyah is sufficient on your behalf and on behalf of your father, your wife, your father’s wife and all the members of your household, one udhiyah in fulfilment of the Sunnah.
End quote from Fataawa al-Lajnah ad-Daa’imah (11/404)
There is nothing wrong with you giving your father the price of the udhiyah so that he can take care of that, and you can intend when doing that to delegate him to do that on your behalf and on behalf of the family.
There is also no reason why you should not give him the price of the udhiyah as a gift, so that he can offer the sacrifice on his own behalf and on behalf of the members of his family, including yourself and those who live with him in the same house. However it is preferable for the udhiyah to be offered in your name, so long as you are the owner of the wealth, and to give your father whatever he needs of that wealth.
But if you live in a house on your own, and your father lives in another house, then it is not prescribed for you to offer the udhiyah on your own behalf and on behalf of your father. So you should offer the sacrifice on your own behalf and on behalf of the members of your household, and your father is not obliged to offer the udhiyah if he does not have enough money to buy it.
If you give him the cost of the udhiyah sometimes, and you yourself offer the udhiyah sometimes, that will be good, in sha Allah, and it will be a way of combining the two good deeds.
See also the answer to question no. 41766.
And Allah knows best.