I would never charge my parents rent. That’s so weird. That being said, my parents would probably buy food or give me a little cash.

Why would you even do that? Do you want to create a problem in your relationship with your father? Because I can't even imagine how that conversation goes. "Dad, it's been great having you here, and it's so nice to know that the house is being looked after while we're gone. But some random family members think you're a mooch and I should charge you rent. So...how's $800/month sound?"

Stop discussing this with family members. It’s not their business.

He’s house-sitting for you. How nice!

The answer is of course no.