I begged my parents for months to get a puppy, but I'm not in the same situation as you. I didn't have any dogs already so it would be the only dog in the household. I basically tried to prove that I was responsible enough and it eventually got through to them.

Also dragging up puppy websites everyday to show my mum all the litters and pointing out every dog we saw when we were out, plus talking non stop about puppies and dogs and training and all sorts.

I did say to my mum at one point when I was getting fed up of them saying no "What would you do if I just brought a puppy home?" and she said "Well secretly I'd love it but don't tell your father.".

Good luck!