I can understand your desire for a little girl.
Try not to think of it as needing to convince your parents about this. Think of it as figuring out with them if this is (or isn't) the best decision for your family right now.
Some Yorkies require an extreme amount of medical care which costs a lot of money. Even if the pup is insured, I'd say $1000+++ is needed for vetting, food, more for grooming, etc. Even routine medical care costs hundreds a year.
Another thing to think about is what will happen to these pups long term. If you go away to college, will your parents keep them?
Your current pup might like a friend, but not all pups get along. They may not be able to be together unsupervised. There are also training issues that commonly come up with Yorkies (potty training, barking, separation anxiety). Lots to think about before adding another one to the family.
Take your time and remember that if your bank account is empty over a $5000 vet bill, your parents will need to be the ones to pay.
Also, reputable breeders (that sell pups at 12+ weeks, do prebreed testing, etc.) take time to screen homes and a lot of times have a waiting list. So I wouldn't plan on getting one before mid August.
Good luck with your search.