Pita has some good advice for you. Medication could really make a huge improvement.

May I ask how old the dog is and what breed?

I foster labs (and other dogs too), but they are very prone to hip problems. This leash works wonders and it is not hard on the person walking the dog. You dont pick up the dog you just alleviate some of the weight on the back legs. I have used it for dog recovering from surgery and senior dogs as well. My young neice has no problems using it so I'm sure it wouldn't be to hard on your parents.

http://www.seniorpetsupplies.com/pro...ash-93-36.html

I don't think it is appropriate to tell your mom to put the dog to sleep. Your mom afterall is taking care of the dog. She will know when the dog is ready to go, and she will know when she is ready to let go. She is not ready to let go right now (and it doesn't sound like the dog is either). Senior dogs are harder to take care of but they can still bring a lot of joy and love to their owners.