Learn the road rules and read the requirements for drivers with license category "A" (motorcycles). Contact parents asking about buying a moped, and ask them to quiz you on the material you have studied.
Arrange to work during the summer holidays and, in confirmation of seriousness, spend money on entertainment, and the purchase of a helmet and other accessories necessary for safety on the road while driving a moped. It is possible that parents will not be able to refuse, seeing tangible evidence that you share their concern. If possible, buy small gifts for mother and father.
Take a moped rented from a friend and show when convenient to parents that you drive it carefully, and follow the rules of the road. It is a good idea if the trip will be useful in helping parents who need to deliver the home products urgently or to buy medicine.
Choose a time to calmly talk to your parents about buying a bike. It is possible that they will expose you to some serious but not impossible conditions. For example, to help with repairs or to take the exam. If these conditions suit you, agree and keep your word, so that parents believe that their son is an adult and is able to comply with the contract.
If you have free space in the garage or shed, you can the money to buy a moped, requiring a serious overhaul. Ask the father permission to carry out repair and Assembly work, and get to work with all the thoroughness, not pretending that the room is really something going on. It is possible that your parents finally realize that you really need a moped, and give it to you without talking.