What if the dog were to die while your folks were on holiday?
I think it may be possible that you are the only one in your family who is able to look at this rationally and UNDERSTAND what is best for the dog. Your Mom is a sloppy sentimentalist who takes no responsibility for the object of her affection and "deep concern." Your Dad is used to being the dog's butler and private nurse. Probably he has more empathy for the dog's situation, because you don't spend night after night cleaning up shit and piss because it is a fun thing to do.
You said, " I just cant stand to see her like this, a shitting pissing whimpering machine."
That is what we all are, isn't it? Read Ecclesiastes. Read Buddha.
"Hi Mom? Fluffy died while you were at Disneyland."
Have you ever attended a sick animal and had it die in front of you?
I have. I promised that little being that I would not let him die in pain, and I failed him because the vet didn't arrive soon enough.
It isn't pretty, or poetic, or painless.
You can give that poor dog a final gift. I think though that the gift will be your secret forever.