What on earth aren't they sure about? Are they worried you'll make a mess? If so, offer to clean up the kitchen too. Are they worried you'll give them all food poisoning? If so, offer to cook vegetarian so that there is less risk of foodborne illness. Are they worried you'll light the house on fire? For goodness sakes, you are a teenager! You aren't going to hurt yourself cooking any more than you'd hurt yourself doing all the other things that teenagers do. Are they worried that none of your dishes will be tasty? Ask them what they like or offer to make them their favorite meals.
Your parents sound really crazy. If I had a kid who wanted to cook over their summer break, I'd let them for sure. Even if they were several years younger than you are.
One compromise might be to ask if you can cook 2 or 3 meals a week for a couple weeks, rather than all 7 in a row. This might set aside any concerns that they have with them having to put up with your food for a whole week at a time.
Another compromise would be if you can do the grocery shopping yourself so that they aren't having to run out and get you weird ingredients for your meal plans.
A final compromise would be to use healthy cookbooks or a food guide when planning out your meals. It might be that they are worried that the food you cook won't be healthy enough to eat for 7 days straight.