The rituals of Hajj and Umrah can only be done on behalf of one person at a time. You cannot do one ‘Umrah on behalf of both of your parents, rather you should do one ‘Umrah on behalf of each of them. It is better if you do ‘Umrah on behalf of your mother first, because the mother has greater rights, unless the ‘Umrah for her is naafil (supererogatory) and the ‘Umrah for your father is obligatory, because he did not do ‘Umrah during his lifetime. In that case you should do the ‘Umrah on behalf of your father first.
It says in Kashshaaf al-Qinaa’ (2/337): It is mustahabb to do Hajj on behalf of one's parents, if they are deceased or unable to do it. Some of them added: If they did not do Hajj. Imam Ahmad stated that he should do it on behalf of his mother first, because she is more entitled to honouring, but the obligatory Hajj on behalf of the father takes precedence over the naafil Hajj of the mother. End quote.
If a person enters ihraam for Hajj or ‘Umrah on behalf of two people, the Hajj counts on his own behalf only.
Al-Shaafa’i (may Allaah have mercy on him) said in al-Umm (2/137): If two men hire a man to do Hajj on behalf of their parents, and he enters ihraam on behalf of both of them, then he invalidates his contract with them and the Hajj is on his own behalf, not on behalf of either of them. Similarly, if he intends to do Hajj on behalf of himself and on their behalf or on behalf of one of them, it is on his own behalf and this contract is invalidated. End quote.
Ibn Qudaamah said in al-Mughni (3/97): If he intends to perform Hajj on behalf of two people and he enters ihraam on behalf of them both, it is on his own behalf and not theirs, because he cannot do it on behalf of them both, and neither of them is more entitled than the other. If he enters ihraam on behalf of himself and someone else, it is on his own behalf, because in the former case he did not intend to include himself, but it was counted as his; in the latter case, the ruling applies even more so. End quote.
Al-Nawawi (may Allaah have mercy on him) said in al-Majmoo’(7/126): Our companions said: If two men hire a man to do Hajj on their behalf, and he enters ihraam on their behalf, then his ihraam becomes his and is not counted on behalf of anyone else, because ihraam cannot be entered on behalf of two people, and neither of them is more entitled than the other. If he enters ihraam on behalf of one of them and on his own behalf at the same time, then his ihraam is on his own behalf, because it is not permissible to enter ihraam on behalf of two, and he is more entitled to his ihraam than anyone else, so it is his. This was stated by al-Shaafa’i in al-Umm, and he was followed by Shaykh Abu Haamid, al-Qaadi Abu’l-Tayyib and our companions.
We ask Allaah to accept good deeds from us and from you.
And Allaah knows best.