I am lucky because I had family support - but many people don't. Your mother, in common with most omnivores, believes that you have to eat meat in order to survive and be healthy. This is because our society has told her that, and that position is constantly being reinforced by advertisements, TV cooking shows, recipes, supermarkets, and by a government who is afraid to tell the truth lest they lose the support (and money) of the large and powerful meat and dairy industries. So please don't blame your parents for their lack of knowledge about nutrition. Don't get upset with them. If you can, laugh along too, and tell them that you will show them that it is possible. Do you have any books about vegetarian nutrition or health or vegetarian recipes that you could show them? if not, you can show them websites like
www.veganhealth.org, or the site of the vegetarian society in your country. You can also tell your parents that there are a lot of meat replacements available, so the truth is that you will not be missing out on much.
It may take a time for them to accept it - after all, you are making a radical dietary change, and by doing so, you are making them question what they do. This makes people feel uncomfortable, and often they have knee-jerk reactions to that, like laughing at you, or shouting at you, or telling you that you are wrong, or trying to make you see how much you will missing. That's normal, just accept it, and they will change: once they see you adapting, and see that you are healthy and happy.
But...make sure you stay healthy - research the nutritional side (you can always ask questions on these boards if you need to know something) and make sure you get enough of the correct nutrients.
We're here for you....good luck with telling your dad - let us know how it goes.