Please don't give up your dog. Whatever it takes to keep her, do it, even if it means no health insurance for her for a while (if that's what you have to do).
Our dogs are family. In your post you even said she is important to your parent's emotional and physical health.
Was talking to a dog park buddy about this very thing yesterday...we both agreed we'd go to a food bank for ourselves before we'd part with our furries. This sounds like a short term challenge. I encourage you to do what you need to do to keep your little companion.