They said that they love us and want to spend more time with us since they are aging. They wanted to sell their home and move in with my husband and I (in our 50's) permanently. I know there's another reason - they need to sell because they are getting financially strapped. We do pay their mortgage. They want to use their money and live a better life and the only way they can do it is to be with us for good. I know that we do get along and we love both of them dearly too, but I do enjoy quiet time with my husband and dog. They also have a dog which is bothersome to my dog. I was left speechless when they told me their idea and my mom now is hurt and disappointed in her daughter. They can rent their home and live with us for 3-4 months if they need the money, but permanent?
So far as the dog is concerned, I've adopted/rescued life long, both dogs and cats. They may not get on to start but I only step in if someone is going to get hurt. They sort it out themselves. They're smarter than we are :)
Curious why your parents are now financially strapped? Did the live beyond their means? If so, that won't change by selling, in fact they will have more money to use depending on the equity in their home. They will need to learn to budget.
If your parents wind up at your front door with a moving van, have your parents pay you room/board, otherwise you will see your expenses double.... they will live the better life, but not you and hubby... and this could go on for many years.
It's a big decision to have someone move in with you indefinitely. It's difficult to anticipate what it entails, but it is very likely you would become their caregivers for the rest of their life. I'd take that into consideration. I'd also figure out if you need to get a written contract for their care. Others here might chime in with info on that.
If I was considering it, I'd do a lot of research, before committing. Doing this without a written contract is risky. If you read at this site, you'll see how these arrangements often go bad. If you do decide to agree to it, I'd let them know that they would have to rehome their dog as that he would not be a good fit. Maybe, that would dissuade them.
For some people these arrangements work out fine. But, if it's just the money, why not get them some financial counseling, a budget and backup plan so they can have some peace of mind. That might allow them the confidence to stay where they are.