What's the right indoor temperature for a senior dog?

My parents have a 12-year-old Corgi-Chihuahua mix who has liver disease. He has been lethargic lately and not eating. My mom thought they were going to have to put him down.

I went to their house yesterday and found the house was 82°F with 65% humidity. The dog was lying on the floor panting. I plugged that into a heat-index calculator and it's a little over 85°. We got into a big argument over why they don't keep their house cooler than that — if nothing else, for the dog's sake. (For context, parents have AC and can afford to run it, they just don't like it.)

Could that indoor temperature be adversely affecting the dog's health? If so, what's the maximum temperature they should keep their place?