My parents have one problematic cat and one problematic dog and they’re ruining their lives

My mom has been doing animal rescue since the 90’s. She is takes pride in taking in unwanted animals and more often than not she’s the one in the rescue group that is asked to “foster” a decrepit/dying/problematic animal. My dad begrudgingly accepts that this is the way they live. They’re now entering their 70’s and retired. This should be the time to travel, relax, go to their social club, play pickleball, etc… but they have this 14 year old Yorkie that was a puppy mill rescue that was never able to be crate trained. She was a breeding dog, so she lived for 8 years in a crate shitting and pissing in it. This Yorkie will only pee and poop in the house - in the living room or kitchen. My mom is too lazy and/or feels like it’s too hopeless to try to train her again. The dog will sometimes poop then mash its poop into the tile and eat some of it. It’s gotten so bad and it smells like a kennel. It’s gotten so bad my Dad makes excuses to leave the house.

Then they have this cat, age 10 or so, that was a feral kitten when they got it. It just walks around the house sneezing. You can’t catch it or it will claw your arm off so there’s no way to take it to the vet. It doesn’t eat treats. Sometimes it looks very ill and it’s struggling to breathe. Other times it’s normal. But there’s snot everywhere. The cat is not interested in going outside at all.

It’s really turned into a zoo and it’s not fair to my dad or my brother (who is disabled and lives with them still). I am trying to convince them to put the cat down. I’m at a loss on what to do with the dog. I know this is my mother’s fault but if she’s not going to put in the time to get the Yorkie trained maybe she should give it to a rescue group that would and at least give it a chance. I know she doesn’t want to look like a hypocrite after judging every owner that gave up their older pets, but I don’t know. Hasn’t the family put up with enough? Hasn’t she made up for it with the other fosters? Do we really have to wait 3-4 more years of this before the dog finally passes? Has anyone else just given up on a pet like this?