What is the ruling on a father, who has been residing in his son's house, remaining there after the son and his family return, which causes embarrassment to the wife, given that there is no other place for the father to go?

Filial piety is an obligation upon children without restriction or condition, and it is not linked to the parents fulfilling their duties towards their children. Therefore, you must show piety towards your father, whether he provides for you or not. You must also provide for his expenses and housing. If you cannot provide him with a separate dwelling, then he should live with you, provided there is no harm in that. It is not permissible to neglect or abandon him.