The sentence with the हैं dropped sounds alright to me.
Funnily enough, due to the dual meanings of चलना, it sounds like the sentence is saying that my father is not functioning (as if he is a robot) but that's also true of the version with हैं.
पिता is treated as a plural noun as a way of showing respect. This is done for all words that refer to people older than you or in a position of authority. This is why it is conjugated with the plural form रहे (rahe).