Praise be to Allah.
If your father is not able to do Hajj himself, because of old age or sickness for which there is no hope of a cure and which makes him unable to do Hajj, then he has to appoint someone else to do Hajj on his behalf, whether he appoints you or someone else.
If you want to do Hajj on his behalf, this is good, so long as you tell him that you are going to do Hajj on his behalf and he gives you permission to do that.
Ibn Qudaamah (may Allah be pleased with him) said: If a person meets the conditions of Hajj being obligatory for him, but he is unable to do it for a reason for which there is no hope that it will disappear, such as old age or sickness for which there is no hope of a cure, or he is too weak and cannot sit firmly on his mount except with unbearable difficulty, or he is too old, etc, if there is someone whom he can appoint to do Hajj on his behalf, and he has enough money to do that, then he must do that. This is the view of Abu Haneefah and al-Shaafa’i. End quote from al-Mughni (3/91).
He also said: It is not permissible to do Hajj and ‘Umrah on behalf of a living person without his permission, regardless of whether it is an obligatory or voluntary Hajj, because it is an act of worship can be done by proxy, so it is not permissible to do it on behalf of an adult of sound mind except with his permission, as is the case with zakaah. In the case of one who is deceased, it is permissible to do it on his behalf without permission, whether it is obligatory or voluntary, because the Prophet (peace and blessings of Allah be upon him) enjoined Hajj on behalf of the deceased, and it is known that permission cannot be sought from the deceased."(Al-Mughni 3/95).
And Allah knows best.