I have recently gotten alot of questions similar to this so i thought i would answer in a post. I can understand why many parents don't want their child to go vegan (and especially not if they have/have had an eating disorder, because unfortuantly... veganism can be equalled to restrictivness for some... if they dont know how/what to eat, or actually allow themselves to eat). If a person isn't educated about veganism, or understands why veganism is a GOOD thing it can be hard for them to accept or understand a persons choice to go vegan. People in the western world grow up learning that we need meat and dairy to survive, and they usually make up huge parts of a persons diet - however most people dont need meat or dairy to survive or to be healthy (Yes, there are some who do, or some who have so many allergies that they couldnt eat a vegan diet and still be healthy. Or people with illnesses who might need meat, but then there are so many more who eat a vegan diet and feel better than ever and actually feel that allergies or illnesses lessen since not eating meat or dairy anymore.)
The best thing you can do is to communicate with your parents, tell them why you want to start eating a vegan diet and maybe ask them if you can eat some vegan meals together during the week so that you can show them that a vegan diet isnt just salad or pasta, but so much more! However, i do think that eating vegetarian while living at home can be a good transitioning step. Many can find it hard to go vegan over a night (for me, i ate soy yoghurt, soy products and oat milk and it was just the chicken and eggs and some dairy i had to cut out, and i did that overnight but it wasnt a drastic change as my body didnt feel good with all that dairy and meat. So it wasnt a huge change i had to make - so i think doing a transition can be a good idea). Maybe you can begin using oat/soy milk at home and soy products instead of dairy and maybe eat some soy products or make bean burgers or things with lentils and chickpeas for lunch.