Praise be to Allah.
So long as the son is well-off, he should give zakaat al-fitr on his own behalf. If his father gives zakaat al-fitr on his behalf there is nothing wrong with that, especially if the father is accustomed to giving zakaat al-fitr on behalf of his children every year even though they are grown up and working. He may like to continue doing what he is used to doing, and it may upset the father if his son tells him “Don’t give zakaat on my behalf.” So the son should give his father the opportunity to give zakaat on his behalf, and he should also give it on his own behalf. Some people consider the father’s continued giving of zakaat on behalf of his children to be a continuation of his connection to his children and a sign that they are still under his authority and care, so the son should let his father do what makes him happy. We ask Allaah to put right the affairs of us all.