That's hard to say. As somebody who has bad knees (and feet, and ankles) and who also owns a dog, I can sympathise to a large extend. A dog is wonderful company and can be a real boon to somebody who is housebound through disability.
That being said, I walk (or limp, or hobble) my dog most mornings in a bushland setting where many local people go with their dogs. A few of those are also walking their neighbours' dogs simply because they love dogs and it gives them a chance to be out there.
Your problem appears to be centered on two words "for free". Not everybody needs to be paid in order to do a favour for a person in need. You may be correct in thinking that you're being taken-advantage of; on the other hand, you may perhaps need to look at your own attitude and wonder if it may be just a little bit mean.
Think about it.