Explain to your parents that raising Tamagotchi can teach you responsibility, dedication, and time management.
If it's the annoying beeping that bothers your parents, tell them that you can turn the sound off, and even pause them if you have homework or something else that you have to do instead.
Also, try saving your own money until you can buy your own Tamagotchi. This shows your parents that you're willing to save up for something that means a lot to you, and proves that it's not something you just want on a whim.
Crying or begging may not be such a good idea; it always back-fired against me when I was little. If you want to prove to your parents that you deserve something you want, I think your best bet is to show that you're serious about it and go about it in a responsible way.