You need to prove to her that it is completely possible to be a healthy vegetarian. Research, read, and read some more! Read up on the matter. Becoming Vegetarian
(Melina and Davis) is great (I'm suprised Fromper hasn't suggested this
) and you have the help of a lot of experienced vegetarians and vegans here on VB
from all walks of life. You need to prove to her that you know what you are doing, that you are going about it the right way and eating a healthy and balanced diet. Parents are picky, I've encountered the same problems with them doubting my nutrition, so I know where you are coming from.
Research will (most likely) help out a lot! It definitely did for me. This way, you know the nuances of the diet, where you are lacking, what vitamins you need to be careful about (ie B12 and omega-3's), and it helps your parents to realize you are completely serious about your decision, not to mention it's kind of impressive when you start talking about iron absorption and yadda yadda.
You might be able to come into an agreement with your parents about the "two meals"- maybe your mom will make dinner, something that can have the meat easily replaced, and you can make the replacement. You could also try making a (balanced) veggie meal for your family once a week or so, proving to them both that you understand your diet, and that vegetarianism isn't "weird" or what have you.
You might also look into being frugal, especially if your mom thinks that you're going to break the bank with "weird" foods. Consider beans (dry). They store for a very long time AND are cheap, versatile, and very nutritious.
It's probably more expensive to be an omni (don't quote me, I haven't looked into it); a vegetarian diet is only as expensive or as hard as you make it.
Sometimes though, parents (and others) still aren't supportive for many reasons. Perhaps they are not ready for you to start growing up and making your own decisions; maybe they feel that you are really doing the right thing (being healthy) and they don't want to abandon the only way of living they have known, hence they are in opposition to it. In this case, you might have to self-support yourself a little. If you have a job (with enough spare money) you might be able to buy food for yourself.
Again, I stress research. And good luck! We're here to help you out.