Parents are most worried about you taking good care of it. To convince them, you need to start with demonstrating you can be responsible. I would suggest keeping a notebook where you write down all of the responsible things you do. For example, do your chores without them asking, then note the date and what you did in your notebook. Do other things around the house without them asking also. Make sure to write everything down in your notebook.
Here are some more ideas you can do and write down in your notebook:
Do your homework, and get good grades.
Be extra good at school so your teachers will let your parents know on your report cards or during meetings.
Clean up any messes you make around the house such as dishes and food wrappers.
Help your parents cook dinner, and do the dishes.
Take out the trash
Make your bed in the morning
Don't leave clothes or towels on the floor
Save your allowance instead of spending it
Make notes of all of the responsible things you've done and once you have a few pages full of notes, then bring it to your parents and show them. Let them know you've been working on being more responsible, you have proof of it, and let them know that you would really like a hamster to take care of and you saved your money to buy it.