Is the father held accountable for anything if he tells his son, "It is forbidden for you to go to the استراحة (chalet/rest-house) for a week," and then the son goes without his knowledge, and is it permissible for the son to atone for this "thing" on behalf of his father?

If a man makes another swear not to do something, and that person disobeys him and does it, then the expiation for the oath is incumbent upon the one who made the oath. It is permissible for someone else to act on behalf of the one who broke the oath in offering the expiation, but the one who broke the oath must be informed of this before it is offered, because intention is a condition for its offering.