Praise be to Allah.
The udhiyah (Eid sacrifice) is a confirmed Sunnah according to the majority of fuqaha’. Some of the scholars are of the view that it is obligatory for the one who can afford it; this is the view of Abu Haneefah and of Ahmad according to one report, and it is the view favoured by Shaykh al-Islam Ibn Taymiyah.
One udhiyah may be offered on behalf of a man and the members of his household, because of the report narrated by at-Tirmidhi (1505) and Ibn Maajah (3147) from ‘Ataa’ ibn Yasaar who said: I asked Abu Ayyoob al-Ansaari: How were the sacrifices (udhiyah) offered at the time of the Messenger of Allah (blessings and peace of Allah be upon him)? He said: A man would sacrifice a sheep on his own behalf and on behalf of the members of his household, and they would eat from it and give some of it away. Classed as saheeh by al-Albaani in Saheeh at-Tirmidhi.
It says in Tuhfat al-Ahwadhi (5/76): This is a clear statement that one sheep is acceptable on behalf of a man and of the members of his household, even if they are many; this is what is correct.
Al-Haafiz ibn al-Qayyim said in Zaad al-Ma‘aad: The teaching of the Prophet (blessings and peace of Allah be upon him) is that the sacrifice of a sheep is valid on behalf of a man and on behalf of the members of his household, even if they are many in number. End quote.
The members of a household include a man’s wife and children, and also a relative if he lives in the same house and is included among those on whom the head of the household spends, or if they contribute to expenses and costs of food and drink.
But with regard to a relative who lives in a separate house, or spends on himself separately, it is not valid to include him in the udhiyah, and it is prescribed for him to offer an udhiyah by himself.
If you and your brother live together, and your father spends on you both, then one udhiyah is acceptable on behalf of both of you, regardless of whether it is bought with the wealth of one of you or you buy it together.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was asked: Is it permissible to offer one udhiyah on behalf of two full brothers who live in one house with their children, and they eat and drink together?
He replied: Yes, that is permissible. It is permissible for the members of a single household, even if they are two families, to offer only one udhiyah, the virtue of which will extend to all of them.
End quote from Fataawa Noor ‘ala ad-Darb.
So long as the udhiyah is shared by both of you, you and your brother, there is nothing wrong with that, even if your father does not pray.
If he is a Muslim, one who prays sometimes and does not pray sometimes, as is the case with many wrongdoing Muslims who are heedless about the prayer, there is no problem and there is the hope that Allah will benefit him thereby and record some of its reward for him.
If he happens to be a disbeliever, then it is you and your brother who are offering the sacrifice, and not your father.
For more information, please see the answer to question no. 285977.
If you and your brother have saved up to pay for the udhiyah from money that your father gave you as a gift or by way of maintenance, and some of it was left over, there is nothing wrong with that, and the udhiyah is valid on behalf of both of you, in sha Allah.
And Allah knows best.