I dropped हैं in the end and it was a mistake. I thought dropping /hona/ in negative sentences is fine. Now I'm confused bout it again..
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.
So why "raho" and not "raha?"
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).
Except we get रही for माँ
Both the feminine singular and plural forms are रही. Eg: मेरी बहनें चल रही हैं (My sisters are walking).
But the pluralisation for showing respect is done for the माँ as well which is why we need to use हैं instead of है in मेरी माँ चल रही हैं