If your parents have a dog, how about offering to take care of him/her for a week or so? You walk him, feed him, etc., and if you do that to their satisfaction, then you get your own dog. (It doesn't prove that you can handle unusual situations like emergency trips to the vet, but it does prove that you can handle the day-to-day requirements of owning a dog.)
And tell your mother that if you don't take the dog, he *will* be taken to an animal shelter and *will* be put to sleep (even if that's not a given - pull on the heartstrings
)