If you don’t think they’ll be supportive, sit them down while you’re all in neutral moods. If you have siblings, have them in on it and on your side if possible. Don’t ambush them, or attack their cooking or anything like that. Explain the health reasons above the “looks” reasons you want to eat better (ie, having more energy, less chance to get a disease that runs in the family, etc). Mention the nastiness that comes in a certain food you want to avoid.
The more people in the family interested in eating healthier, the more healthy food will be bought. Get everyone involved!
Ask to help make the grocery list or come to the grocery store.
Offer to cook once or twice a week as long as you can pick what the food is. This is a great way to get more healthy food in the house (leftover ingredients and leftover meals) and it gets your family trying new kinds of foods. They’ll learn that healthy food can be delicious too!
Research and explain why you want one food instead of another. Your parents may not know why brown rice is better than white - or why whole wheat is better than white bread.
Start small. Don’t demand 100 bucks in health food right off the bat. They don’t know that you’re gonna eat it all! Start off with healthier alternatives (ie, light string cheese instead of full fat, organic apples instead of regular, quinoa instead of rice, whole wheat bread instead of white) and go from there. You’re more likely to stick with the healthy lifestyle this way anyway.
If money is an issue, try to find coupons online or in the mail for the things you want. Often if you write to the company that makes the product you like, they’ll send coupons or free samples back.
Be prepared to get less than you wanted. Remember that they’re still your parents, and its still their house and their money. It may not make sense to buy unhealthy food when there healthy options available, but in the end its up to them. If the answer is no, or way less than you wanted, try to ask for a small part of a cabinet or pantry. When you have some extra cash (or when you get a job) stash your food there. Keep it easily accessible and away from everyone else food so it doesn’t get lost behind the junk.