Suggest to your parents that you would join a local cycling club and learn the ropes with club members. You would only ride in groups until you have mastered all the techniques and rules of the road. Hopefully this would make your parents see you are going to go about it in the best manner and that the comfort of you being with a group should set them at ease. Most clubs have very reasonable membership rates. You could try and get both or one of your parents to join with you.