It is not permissible for the one who has entered ihram for Hajj or ‘Umrah to exit ihram or stop doing Hajj or ‘Umrah before completing it, because Allah, may He be exalted, says (interpretation of the meaning):
“And perform properly (i.e. all the ceremonies according to the ways of Prophet Muhammad صلى الله عليه وسلم), the Hajj and ‘Umrah (i.e. the pilgrimage to Makkah) for Allaah”
If your father is able to go back to Makkah and complete the ‘Umrah for which he entered ihram, then it is obligatory for him to do that, and he cannot exit that ihram except by completing it, so long as he is able to do that.
But if there is any genuine excuse that is preventing him from completing the rituals, then in this case he is muhsaran (prevented), so he should offer a sacrifice in the place where he could no longer carry on, or slaughter it in Makkah and distribute its meat to the poor of the Haram, because Allah, may He be exalted, says (interpretation of the meaning):
“But if you are prevented (from completing them), sacrifice a Hady (animal, i.e. a sheep, a cow, or a camel) such as you can afford…”
Then after sacrificing an animal, he should shave his head or cut his hair. Then he will have exited the ihram for ‘Umrah.
As your father was not able to complete his ‘Umrah and he went back to your city because he was sick and is elderly, then in this case he should send someone to Makkah who can slaughter the sacrifice for him there and distribute its meat to the poor of the Haram, then he should shave his head or cut his hair. And he has to avoid the things that are forbidden whilst in ihram until he has done that, because he is still in a state of ihram for ‘Umrah until he does that which Allah has commanded the one who is prevented (from completing Hajj or ‘Umrah) to do.
See also the answer to question number 37995.
If he stipulated when entering ihram, “Allahumma mahalli haythu habastani (O Allah, my place of exiting ihram will be where You caused me to be prevented (from continuing)),” then he may exit ihram for ‘Umrah and he does not have to offer a sacrifice or shave his head or do anything else.
With regard to the ‘Umrah that your brother did on behalf of your father, this is a separate ‘Umrah and is acceptable on your father’s behalf if he is unable to do ‘Umrah because he is elderly or is sick and has no hope of recovery.
But if he is able to do ‘Umrah for himself, or there is hope of recovery from his sickness, then in this case it is not valid to do ‘Umrah on his behalf.
Please see the answer to question no. 41732.
When you went back to Makkah you should have completed the ‘umrah for which you entered ihram with your father, and you should not have started ‘umrah from scratch, because you were still in ihram. Hence your second ihram is regarded as superfluous, and your tawaaf, saa‘i and shaving your head are regarded as a completion of the first ‘umrah.
If you did any of the forbidden actions after you returned from Makkah, until you went back to complete your ‘umrah, then you have to offer a fidyah for each of the forbidden actions that you committed, unless you did it out of ignorance, in which case it is pardoned.
And Allah knows best.