Is it permissible to fast on behalf of a father who passed away at the hand of someone, and who was obligated to fast for two months, knowing that he was ill and could not endure fasting?

It is not legislated for you to fast on behalf of your father as long as he is alive. The scholars have unanimously agreed that it is not permissible for anyone to fast an obligatory fast on behalf of another person during their lifetime. Even if one were to fast on his behalf, it would not suffice him nor would it relieve him of the obligation. If your father is unable to fast due to illness or the like, the obligation of fasting remains binding upon him until he is able to perform it. Some jurists are of the opinion that it then transitions to feeding the poor.