There are several different ways to get the protein out of animal meat without actually eating meat. Nuts, fruits, and grains have plenty of protein in them. People can also get it (since we're talking about vegetarians and not vegans) from milk, eggs, cheeses, and yogurt. There really isn't any danger of a protein deficiency from a vegetarian diet.
Children are capable of having strong values as well. I don't think it's right to demean that just because you're a parent and older.
A parent can decide to put forth an effort and find a way to allow the child to do what he or she feels he needs to do, while keeping him healthy. All the parent has to do is do some research to find other means for proteins and other neutrients, and tell the child what being a vegetarian entails, and explain what they'll have to eat in place of meat, so they can stay healthy.
It's a parents responsibility to take care of their children. Meat has protein and other things that are necessary for someones growth. The child can make their own choices when they grow up and move out. But until then, the parents choose.
You can eat biological stuff like that the animals had a good life, but those kids really need their proteins. I know two girls who have been a vegetarian since they were 9 or so and they are really short and pale, while their families are tall and tanned(no joke).
Also if that's the case they can't eat eggs, cheese and stuff either. Also then they can't wear leather and stuff.