Good for you for wanting to become vegetarian! It might not be a bad idea to pick a time to set aside and talk to your parents about why you want to become vegetarian, what it means to you. Don't make it just a conversation in passing, but really let them know why and how important this is to you. Use references from the library if you need to. Also, take the time to research what to eat so that you can show your parents this information. It will make them less worried for your health and so on if they see that there are so many healthy foods to eat as a vegetarian. Here are some resources for you to look into:
If you hide your vegetarianism from your parents and then they find out, they will only be more upset and think you have something to hide as a vegetarian. It would be best to be up front. You could also offer to pitch in with dishes and chores and cook vegetarian meals for your family on occasion. You could ask to come along to the grocery store and help pick out the foods you would eat.
Here are some vegetarian foods:
if you can eat sunflower butter (from sunflower seeds and not part of the nut/peanut butter families) it is really good stuff! It is made especially for people with peanut and nut allergies. Most larger chain grocery stores carry it.
beans...kidney beans, white beans, black beans, chickpeas, limas etc. Add them to salads, mash and make sandwiches with them, make bean and vegetable soups and so on.
grains like oats, rice, pasta, bulgur, barley, cream of wheat, cornmeal. They make great hot cereals with fresh or dried fruits or use them as a base for vegetables and beans and sauces to add to them.
snack on fresh fruits, dates, hummus and raw veggies or tortillas or on toast; homemade granola or muffins; make some tofu pudding (blend tofu with cocoa powder, maple syrup or another sweetener, a banana, and a splash of plant milk;
You can make tacos with refried beans (canned fat free refried beans do not have lard in them but full fat refried beans usually do) or with crumbled tempeh and add taco seasonings tomatoes and lettuce. You can eat spaghetti with tomato sauces simmered with vegetables (mushrooms, onion, green peppers, zucchini etc) and with red lentils cooked and mixed in for a protein boost.
Like you I was very tiny until I reached sixteen, and then filled out. But I became anorexic in my thirties and still suffer the aftermath of six years of being severely underweight. I have osteoporosis and am being treated for it, and have to sit on a donut cushion now because my tailbone is bruising from the gravity and weight sitting places on it (and I am not even underweight anymore). That is how bad my spine is. Not to scare you, but please make sure to take care of yourself and get enough to eat. Your growing bones need all the help they can get.