First off your parents do not know best. Eating animal products is incredibly bad for you. We are the ONLY species on the planet that consumes dairy after infancy. We are also the only species that eats other species's dairy. Our bodies can not handle it. Milk is meant for newborns only.
How do cows or elephants get protein? Through vegetables. Eat rice, beans, tofu, tempeh, seiten, flours, nuts, peanut butter. Protein is in many things.
Vegan isn't a diet. You should know that. Vegan is a lifestyle. Vegans own nothing and eat nothing that came from an animal. That means no dairy, no eggs, no honey, no wool, no leather, no fur, no gelatin, and no glycerine. They also don't use items that were tested on animals. Like cosmetics, shampoos and dyes. Not all are tested on animals but many are. The actual list of what animal products, by-products and parts are in is exhaustive and vile.
A Strict Vegetarian though is what you might want to try first. Use it as a stepping stone. A Strict Vegetarian doesn't eat eggs or dairy. What lies between that and vegan is your choice. You could eat honey or not. Wear leather or not. While I don't think you should use animals, that's my opinion and it's your choice.
Strict Vegetarian's are FAR healthier than Omnivores. People that don't eat any animal products have a 5% chance of having a heart attack. People that are omnivores have a 50% chance of having a heart attack.
You are less likely to become overweight which lessens your risk of diabetes.
If you don't eat organic meats you're pumping your body full of hormones and antibiotics. That is very bad.
If you eat organic meats, while avoiding the hormones and antibiotics you are still consuming meat. Therefore you still have a 50% chance of heart attack.
Eating meat puts you more at risk for prostate cancer (if you're male) or ovarian (if you're female). Processed meats, like bologna, hot-dogs etc, are even worse than regular meats.
You have to remember that us humans are designed to be herbivores. Yes, we are. We have made ourselves into omnivores and it's fattening us, poisoning us and killing us. Comparatively less people died of heart attacks 150 years ago, than they do today. Yes we have longer life spans, however that is not due to animal consumption. Diseases have increased since we started becoming a more meat-centric diet. As has our medical technology, which one could strongly argue is why we live as long as we do.
So as you can see not eating meat is better for you all around. Cutting out all animal products is the best option for you. Take your time though. What you have is an addiction to animal products. You don't see how life can be without them, it's a foreign concept. Like with all addictions it takes will-power, perseverance and a gradual mind-shift.
I recommend cutting out dairy first. Our bodies absolutely cannot handle dairy. Then, when you're comfortable without dairy drop eggs. So on and so forth. Set your own pace ultimately find what works for you.
I am a Vegan. I was a vegetarian for 3 years before I made the transition. I dropped dairy then eggs. I've been Vegan now for a little over a year. I'm healthy and still kicking.
Remember, yours is the healthier lifestyle already. It's your parents that have the unhealthy eating habits. What they are eating is surely killing them. Tell them that.