Is it permissible to refuse someone's request for a ride to a place that is not on my way, even if they are elderly, and would I be considered rude if I refused?

There is no religious obligation to carry the mentioned man and take him to his home, especially since his daughter is present and capable of carrying him, and refusing to do so is not considered rudeness. Similarly, there is no obligation to obey the mother in her command to carry him, as there is no benefit for her in it, and due to the hardship it would cause the son. Nevertheless, taking the initiative to carry him, if possible, is considered an act of filial piety towards the mother and a means of gaining her satisfaction, in addition to complying with the Sharia's encouragement of showing sympathy and helping those in need.