Is it permissible for my father to perform tayammum due to difficulty in moving and walking because of knee osteoarthritis and a skin disease affecting his toenails?

Difficulty in walking or a skin disease in the toenails does not permit tayammum (definition). Rather, your father must purify himself with water as much as he is able, even if it is with the help of others, unless it is proven that water would harm him, based on a doctor's advice or personal experience. Tayammum is not permissible if one is able to use water, and the fear that is considered valid is that which is based on a real cause.