I'll elaborate, but here's a couple of quick pointers:
- Be honest, but don't criticize.
- Stay positive.
- Be prepared to cook (and be ready to learn how to cook, if you don't know how yet).
- Have some facts ready in case you need them.
Be honest with them, but also don't sound like you're criticizing them. When you say buzzwords like "disgusted," a lot of meat eaters will think you're calling them disgusting, even though you're referring to the treatment of the animals. Instead of saying anything negative about a diet that includes meat, have some positive reasons to go vegan instead--it'll make it much more likely that they'll be receptive and supportive.
Stay positive and don't use negative phrases. For example, instead of saying "meat is so gross," say something like "I want to improve my health by eating more veggies" or "I want to learn more about the variety of veggies/fruits I can eat" or "I feel that the vegan diet is what's best suited for me." Keeping it positive and pro-vegan instead of negative and anti-meat will make it go much better.
Be ready to cook. I work with a gentleman whose 16-year-old daughter went vegan a few months ago. Him, his other two daughters, and his wife didn't want to change their diet, but the 16-year-old daughter told him how important it is to her to become vegan. She kept it very positive, and said that she wanted to work on being as healthy as possible and eating lots of veggies. He was supportive, but told her that no one else would go vegan or vegetarian and she was responsible for maintaining her diet. So, she started learning how to cook. He buys the groceries for her (since she doesn't have an income), but she does all her own meal planning and cooking. If your parents love to cook and always cook with meat, it can help if you're prepared to start cooking for yourself (and it might even help inspire your parents to eat less meat if they see you making delicious meat-free foods).
My coworker's daughter has learned so much not only about being vegan, but also about cooking. He told me the other day that he's really impressed with how much she's grown her cooking skills. He thought she would stop being vegan and get tired of cooking; he also thought she'd only be eating rice and beans, but she's been cooking up tons of variety. He says it's been incredible to see her so empowered by learning how to cook and take control over that portion of her life.
If you don't know how to cook or don't know what recipes to make, just think of the internet as your menu! In the mood for a burger? Google "vegan burger recipes." Want some mac-n-cheese? Google "vegan mac n cheese recipes." Want something easy to make? Google "easy vegan recipes." There are hundreds of thousands of recipes online, and use this forum as a resource as well. If you don't know how to cook yet, you can also watch scores of how-to videos on YouTube that will guide you. (And you are also more than welcome to ask me any time--I LOVE to cook and have dozens of recipes I've tried, tweaked, and created.)
Have some facts ready. Your parents are your parents, and they'll probably be concerned about your health and wellbeing. There are a lot of misconceptions about the vegan diet, such as all vegans are protein deficient (not true) or malnourished (also not true). Check out some resources (this forum is a great place to start) on where vegans get protein, common misconceptions about veganism, the risks and rewards of being vegan, etc. If you have some good information, you can address any of your parents' concerns in a calm and informative manner.
Best of luck, and if you need anything please just ask!!