I am about 5 years older than you but I too was in the same position. I convinced my parents by being responsible. I now have 3 dogs to show for it.

I did extra chores around the house and made it so that my parents didn't have to do one thing doggie related. When my dog had to go outside, eat, walk, etc. I was the one to do it. And yes, picking up doggie wastes was my responsibility, too. I saved my money for the adoption fee.

Ultimately, it is your parent's decision as they are the ones who have to support financially. I was able to get a third dog and make many decisions when it comes to my dogs because I now have an income and pay for them.

Generally, if they say no.. there's a very good reason behind that choice.