An interesting thing about iron on a vegetarian/vegan diet is that the absorption of non-heam iron (the iron you get from plants) is increased along with vitamin C (which is generally higher in veg diets than meat diets).
That means, it's generally easier to gain iron from your food as a vegetarian (meaning, to reach recommended levels, you don't actually need to consume AS MUCH)!
There are heaps of plant-based foods high in iron such as fortified breakfast cereals, wholemeal bread, dried fruit (e.g. apricots, prunes, raisins), green leafy vegetables, beans and pulses, molasses, nuts and seeds (almonds, pumpkin seeds, sesame seeds), tofu. Hmm... that's more sources of iron than you'll find in animal products I think!
My blood tests since being vegan have shown that I have a higher level of iron than the average meat-eater (and I don't take iron supplements, never have!)
Some people are naturally more susceptible to iron deficiencies than others. And in fact, anaemia is no more common in vegans and vegetarians than in meat eaters. So if you're predisposed to an iron deficiency, you'll probably need to up your sources or iron or take suppliments, regardless of what diet you follow. But eating veg will not make this any more difficult.
It really is a myth that we need meat as girls to gain enough iron from our diets. This myth has been used by the meat industry to get us to buy more artery-clogging animal flesh. Millions of people around the world survive healthfully without meat and have done so for countless generations.
In fact, the average life expectancy for a vegetarian is GREATER than the average life expectancy for a meat-eater. If I were you I'd mention that to your parents and say that you're interested in living a long and healthy life -- and the statistics show that vegetarians live longer!
Good luck on your journey. It can be very hard when your family doesn't understand. But with your help they might learn a few things about how healthy it is to not eat animals.
Check this out for more info:
http://whyveg.com/nutrition.php