Hey there. I had a lot of trouble with my parents regarding this as well, especially with veganism.
I researched nutrition, learned how to cook some super healthy, super tasty veg recipes and sat my parents down to talk to them in as calm and mature a manner as I could (parents LOVE it when kids sit them down and talk to them like that. It makes them feel you're rational, level headed and grown up enough for your points to be valid). It takes a bit of persuasion, and a willingness to put in the effort; start cooking your own dinner at night, offer to go food shopping with them and use that to influence their purchases, et cetera.
It takes persistence so stay strong and good luck.
Great advice.
Showing your parents that you're really willing to go veg, for example, by making your own meals or helping to cook a veg-friendly family meal and sort of let them see for themselves that going veg won't take anything away from you. After all, we all have choices so also let them see that too, because you're not going to be 14 forever