What to do with parents Cat?

For context

My parents have a 13 year old cat that they feel they can no longer care for. The cat isn’t in poor health or anything. I found out they asked their veterinarian if they’d just put the cat down ; but the vet said no as the animal is too healthy.

I am unable to take the cat myself, or I would in a heartbeat.

In Moncton area, what are the best options for my parents to give up this cat to a good home, without just taking it out to the woods and abandoning it, or putting it down unnecessarily?

Any advice would be appreciated? Previous family pets always just passed naturally or were put down at the end of long healthy lives. So this is the first time any of us have had to think about giving one up.

Thanks, I’m lost as to what to do.

Edit:

Thanks everyone for the suggestions! Especially My Grandfather’s Cat!

Yes, I know my parents suck — that’s why I asked here for help