Freddy, you are between a rock and a hard place. I can feel the pain and anguish you are experiencing over the health of the dog. You are right, you will cause irreparable damage between your neighbours and your family if you take any direct action without their consent. Perhaps you may suggest to the dog owners that you noticed the pain the dog is experiencing and that you would help them by paying for a vet. This way the final decision would be up to the dog owner. You then demonstrate you are willing to help and their consent or rejection of help is entirely up to then. If help is offered with a smiling face then nobody loses face.
Regards