My dog was injured by a 11 yr old in my home, the dog suffered neck trauma and an inflamed spine.. he cannot walk or stand and is on steroids, his condition is getting better but, it is a long process so far my medical bill is 4200.00 which included hospital stay, MRI, medications. The child parents have no plans to accept any responsibilities. What do I do legally and do I have a case?

What are the facts? What exactly happened?

You certainly can sue them, whether you have a case or not is hard to tell. Do you have proof that the kid deliberately tried to injure the dog? Even if it was rough play and the child didn't understand what he was doing you may recover a share of the costs.