Ask your parents if they'll pay you for simple chores like washing the dishes, vacuuming, or dusting. You could even set up a weekly “rate” for these tasks. Consider what household chores your parents really hate and offer to do those each week for a rate that they think is fair.
If you already have weekly or daily chores, ask your parents for chores beyond the usual. Explain to them that you'd like to start saving money. These chores might be ones that are monthly instead of weekly, such as mowing the lawn, raking leaves, or washing the cars. Go beyond the normal chores by suggesting more time consuming projects that only need to be done once but that might take a couple days. Ask your parents if they will pay you to organize the garage or attic, clean the gutters or baseboards, or plant a flower bed. You could also ask your parents for an allowance increase if you complete an additional chore or project on a weekly, bi-weekly, or monthly basis. For instance, suggest mowing the lawn every weekend or every other weekend for an increased allowance.