Praise be to Allah.
The scholars – apart from the Hanafis – are agreed that a man’s udhiyah is acceptable on his own behalf and on behalf of the members of his household, as it is a communal Sunnah, because of the hadeeth of Abu Ayyoob al-Ansaari (may Allah be pleased with him), who was asked: How was the udhiyah done at the time of the Messenger of Allah (blessings and peace of Allah be upon him)? He said: A man would offer a sheep on his own behalf and on behalf of the members of his household, and they would eat from it and give some to others until the people began to compete and it became as you see it now.
Narrated by at-Tirmidhi (1505); he said: It is hasan saheeh.
The scholars differed concerning the definition of the household for which one udhiyah is sufficient; there are four views:
1. Those concerning whom certain conditions are met: the one who is offering the sacrifice spends on their maintenance, they are related to him, and they live with him. This is the Maaliki view.
It says in at-Taaj wa’l-Ikleel (4/364), which is a Maaliki book:
If he lives with him, is related to him and he spends on him, even if it is voluntary, then that is permissible for three reasons: being related, living in the same accommodation and spending on him. End quote.
2. Those on whose maintenance one person spends. This is the view of some of the later Shaafa‘is.
3. All the relatives of the one who is offering the sacrifice, even if he does not spend on their maintenance
4. Those who live with the one who wants to offer the sacrifice, even if they are not his relatives. This was the view of al-Khateeb ash-Sharbeeni, ash-Shihaab ar-Ramli, and al-Tablaawi among the later Shaafa‘is, but this was regarded as unlikely to be correct by al-‘Allaamah Ibn Hajar al-Haytami (may Allah have mercy on him).
Ash-Shihaab ar-Ramli (may Allah have mercy on him) was asked:
Can the Sunnah of offering the udhiyah be done on behalf of a group of people who live in one house, but they are not related to one another, if one of them offers an udhiyah?
Yes it can. Some of the later scholars said that this is applicable if one of them is spending on the rest. End quote from Fataawa ar-Ramli (4/67)
Ibn Hajar al-Haytami (may Allah have mercy on him) said:
It may be that what is meant is his relatives, both men and women.
Or it may be that what is meant by members of a household here is those who maintenance is spent on by one person, even if that is voluntary.
The words of Abu Ayyoob, “A man would offer a sheep on his own behalf and on behalf of the members of his household” may be interpreted in both ways.
Or it may be that what is meant is the apparent meaning, which is that they live in one house and share the amenities, even if they are not related to one another. Some of the scholars stated this definitively, but it is far-fetched.
End quote from Tuhfat al-Muhtaaj (9/345)
To sum up, it is prescribed for the older son who lives in a house on his own, separate from his father, to offer a sacrifice on his own behalf, and his father’s sacrifice is not valid for him because the son – now – is not one of the members of his father’s household, rather he is the head of a separate household.
But if the son voluntarily helps his father with the price of the udhiyah, then he will attain the reward, in sha Allah, but it will be the reward of voluntary charity, not the reward of udhiyah.
See the answer to question no. 41766
And Allah knows best.