Convince your parents that you are mature enough. Do all of your chores without complaining, surprise them with a favor, and don't pick any fights with siblings.
Don't be demanding. Sit down with your parents and show them what the game is, how it works, and the parents' guide to the game. Some parents may change their minds if they know what's going on, and may be happy to see that providing a membership can keep you occupied for longer. Alternatively, offer to pay for a RuneScape membership card.
Offer to pay for it yourself if they still say no. Promise to them that you won't spend every waking moment on the computer. Make a promise sheet for you and your parents to sign that includes which hours or how much time you are allowed to play RuneScape if membership is allowed. Another option is shorten or limit the membership by a few months, for example a six month limit. You can even spare out your monthly allowance in exchange for membership.
Do something for your parents. Rub their backs. Cook a meal for them. Do your chores properly and promptly. Wait until they are in a good mood, then ask them.
Wait until you get your report card. Try to convince them that your grades are good, and you can still keep up with your grades while having some fun on the computer.
Go sit beside your parents and play RuneScape. Go look for members objects and each time you see some say, "Oh no, you must be a member for this." Do this until they come to you.
Look for neighbors or anyone in your relative and friends who might need some help, or part-time job during summer break. Let them know you want to do some simple work, like walking the dog, painting the fence for 1 - 3 months of membership in RuneScape.