maybe the best thing is to ask to go shopping with you mum, pick out some vegan friendly food (you can say the product was recommended by friends) and then ask to cook a meal on your own.
when they're eating it (if they enjoy it), you can say you've made a decision to go vegan - or maybe use the word vegetarian (it doesn't sound so full on

).
Show them at this decision will not be a problem at all for them, show them you can easily change the the family's food to something vegan friendly for yourself - like when having pasta, ask your mum if you can take out some pasta sauce before she adds the meat. I'm sure if you show them that it's healthy, your MATURE decision behind it and MOST IMPORTANTLY showing that it will not be a problem for them.
we're all here for you, be sure to tell us how it goes. good luck! (: