My solution is simple. Move out and get your own dog and prove your parents wrong by actions rather than mere words.

If you can't afford to live independently yet, get a room mate or two. Hit up Craigslist and other related classified sites to find some roommates if you don't know anybody who would bunk with you.

I highly doubt you will convince them while under their roof in any reasonable amount of time so it's best to wait until you move out and not get your hopes up.

You're over 18 so they can't legally keep you from moving out. If they try to tell you not to and you feel you're ready, do it anyways.