Hi. I avoid the term ‘owner’, for similar reasons. I would not be comfortable with mum and dad, but that’s just my preference. However, I certainly understand people feeling that the pets are part of their family.
The term ‘house’ doesn’t truly represent the variety of homes we sit. Apart from what might be seen as traditional houses of various shapes and sizes, there are many sits done in apartments and condominiums. Aside from those, I have seen sitters mention yachts, houseboats, and so much more. I realize it’s difficult to come up with all-encompassing terms or words. As to ‘house owner’ - what about the homes where the occupant is a renter? They are not the owners at all. The plot thickens … 
For a parallel post about word choice, you might want to look at Housesitters or Petsitters?