Is it permissible to perform Umrah on behalf of oneself and one's parent with a single intention at the Miqat?

If you intend to perform 'Umrah (definition) on behalf of your father, the prescribed way is to say: "Labbayka 'an [name of person]" (Here I am, at Your service, on behalf of [name]). However, what you did does not invalidate it, as deviating from the prescribed way does not nullify it. Sheikh Ibn 'Uthaymeen was asked about what a person performing Hajj on behalf of another should say at the beginning of ihram (definition). He replied that one should say: "Labbayka 'an [name of person]," citing the hadith (definition) of Ibn 'Abbas, where the Prophet, peace and blessings be upon him, heard a man say: "Labbayka 'an Shubrumah." He asked, "Who is Shubrumah?" The man replied, "He is my brother or a relative of mine." The Prophet asked, "Have you performed Hajj for yourself?" The man said, "No." The Prophet said, "Perform Hajj for yourself first, then perform Hajj for Shubrumah." If you forget the name, say: "Labbayka 'an man a'tani wikalah fi al-Hajj" (Here I am, at Your service, on behalf of the one who authorized me for Hajj). It is not necessary to repeat this formula during the tawaf (definition), sa'i, or other rituals, as long as you made that intention from the beginning of your ihram. Performing 'Umrah on behalf of one's father is sufficient if he is deceased, or if he is alive but incapacitated by an illness from which recovery is not expected, provided he grants you permission to act on his behalf.