Is it permissible to offer the fidyah (expiatory payment) for fasting on behalf of a chronically ill father who is financially incapable, or to give him money for that purpose, knowing that the questioner is the family's sole provider?

Your father, who suffers from a chronic illness from which there is no hope of recovery, must feed a poor person for each day if he is able to do so. If he does not have money, then there is nothing upon him. You may pay the fidyah (definition) on his behalf with his permission, and there is no objection to giving him the money to buy the food with it. Your spending on your father does not obligate you to pay expiation on his behalf; expiation is an act of worship related to the person himself, and it is not in the category of expenditures. You may pay the fidyah on behalf of your father with his permission, as an act of charity and filial piety towards him.