Parents can almost always say this when you ask for a pet - "You need to prove you can handle the job." My parents have always said that to me. And normally, we don't listen and ask more. But if you turn out to be listening and you are now willing to do the challenge, here are steps you should do.
Clean your room -
Your room is just like your cage. If you are planning to get a hamster, gerbil, guinea pig, or fish, cage/tank cleaning is a must. They spend pretty much all their time in there. Your room, most of you probably don't spend so much time in, but if it's clean and nice, your parents will start assuming that you will do a good cleaning job. Even with dogs and cats - you need to clean up what they make.. This isn't only a one time cleaning. You need to have it clean for like, a month, or keep cleaning it for that long.
Follow some of Bea's tips -
Her forum is a sticky in pets. Use those tips to find out which animal you want and how to care for it.
Make food for the family, or get food -
will be the one getting the pet food when you get your pet and you will be feeding it. Don't forget about water, also. Your family needs food and water, so for dinner some nights, go out, buy some food, or make a nice dinner for them. Serve it.
Pets are serious. With or without your parents knowing, you should practice care for your pet before you get it. Practice serving your family, practice cleaning the cage, practice taking walks and make a checkpoint on when you go back home. Everything you can.
That's it. I'm just going to say good luck on getting your pet. These steps may not get you a pet, but they will sure help you in the future. Good luck, and if you get one, congrats.