Educating your parents is the key.
Let them know why you want to be vegan, that you no longer wish to participate in violence towards other animals.
Research about how you can get all the protein, iron, calcium, etc, you need on a vegan diet. My parents made me do this before they allowed me to go vegan. Parents just want the best for their children, so they might worry you won't get enough vitamins and minerals on the vegan diet. Just like an omnivorous diet, it's possible to be healthy or unhealthy on a vegan diet. For example, if you have a diet consisting of white bread and chips, although it might be vegan, you won't be a very healthy vegan. This will lead your parents and other people to believe that veganism isn't healthy. So a well-planned vegan diet is important. A healthy vegan diet can provide you with the essential vitamins and minerals you need - the only thing it would be best to supplement for is Vitamin B12. The best absorbed form of B12 is called Methylcobalamin - so keep an eye out for that name when you're choosing a supplement.
If they're still convinced that you need animal products to grow big and healthy - have them read The China Study! See if it's available at your local library. After reading this book, my parents realised how bad animal products can be to our health and finally allowed my two youngest siblings to go vegan. Before, they hadn't been allowed to go vegan because they thought they needed to drink cow's milk for strong bones. Actually it's quite the opposite!
Your parents might also be worrying about cooking. Having to prepare one vegan meal for you and a meat meal for the rest of the family. There are a few things you could suggest to them. You could have what everyone else is having, replacing the non-vegan food for vegan. Such as, if they're having rice, spinach, carrots and meat, you could replace the meat with lentils, chickpeas or veggie patties. In my early days of being vegan, this is what I did.
You could show them how to 'veganise' family favourites, such as Pizza, Mac n' “Cheese”, Spaghetti and (Tofu, Lentil or Chickpea) balls, etc. Searching online for vegan versions of your favourite recipes will usually bring up at least one recipe you can try. My parents loved to find new ways to cook family favourites that everyone could enjoy. Gradually, they started cooking less of the 'meaty' meals and more vegan meals for the family. It was just easier for them.
If they're not willing to cook vegan for you, find some easy recipes online and see if you can cook your own meals. Once you get the hang of it, you could suggest cooking a vegan meal for the family every week or so. Show them that vegan meals can be just as delicious as their non-vegan meals!
I have some easy recipes here that you can try. More will be added soon!
And lastly, they might be worried that it would be too expensive. Sure, it might be costly if you get a lot of specialty foods such as fake meat products and vegan cheese, however the foods that should make up most of your diet - vegetables, fruits, grains, legumes, and nuts, are pretty cheap. If my family of 7 on a tight budget can easily buy vegan - anyone can! My parents even found that their shopping bill went down a bit after they stopped buying animal products.
I hope these tips will help you show your parents that it's perfectly healthy to be a vegan, it's delicious, it's not expensive, and I hope they realise that if their child wants to stop harming others - they should encourage that!