As a Scout Leader, I've seen this a lot of times. Typically the parent of a boy of about 13/14 years old expresses astonishment (and sometimes embarrassment) at the amount their son will suddenly eat, if given the opportunity.
I don't want to be sexist here, obviously in many families it is the father who does the cooking, but in my experience it's always been mums rather than dads who find the amount that their adolescent sons eat astonishing, and sometimes equate it with greediness.
I think the reason for the astonishment - and, sometimes, simple lack of belief that they really are that hungry - is that boys go quickly from (in pre-adolescence) needing fewer calories than an adult woman to (in mid to late adolescence) needing more - sometimes a lot more.
!Table 2-3. estimated calorie needs per day by age, gender, and Physical
Source: Dietary Guidelines for Americans 2010, U.S. Department of Agriculture
/ U.S. Department of Health and Human Services.
Other sources which basically confirm the same thing: NHS1 (children), NHS2 (teens)
So, depending on your level of physical activity, as a 16-year-old male you now need from, if you do no physical activity at all, 2000 calories per day (about the same as your mum, maybe a little bit more) up to, if you're very active, 3200 calories per day (suddenly way more than your mum - more than anyone else of any age).
What to do? Here's some suggestions:
Talk properly to your parents. As you're their oldest they may simply not have realised that your calorific requirement is now likely to be so much higher than theirs.
Fill up on cheap foods which are easy to cook extra quantities of. Here in the UK we have 'Smash' (instant mashed potato) - i.e. carbohydrate which you make with hot water and powder - I'm not sure what the equivalent is in other countries. But filling up with potatoes or rice is generally a pretty decent idea. Don't expect to get twice as much of the expensive-and-delicious part of the meal as everyone else - the rest of the family are likely to perceive that as unfair. "Mum, can I have an extra baked potato?" is likely to go down a lot better than "Mum, can I have an extra joint of beef?"
Don't eat crap. It'll be a lot easier to convince your parents that you're not simply being greedy if you're asking for foods which are healthy, balanced etc. Make sure you're getting your 5 fruit and veg per day, and all that.
Learn to cook. It's my personal opinion that, at 16yo, if you're not cooking dinner for the whole family at least once a week then you're not playing your part in the family as a young adult. Your family may have a different view, but I'm betting that if you combine your request for more food with an offer to do more of the shopping and cooking (for the whole family, not just for yourself) then it's more likely to be taken seriously. As in, "hey, mum, dad, two issues I'd like to talk about, 1: I need more calories per day than you do, and 2: put your feet up and have a cuppa while I take responsibility for buying and cooking dinner for the whole family at least once a week".
Bear in mind (and raise in your conversation with your parents) that calorific requirement varies very widely depending on your build and your level of physical activity. So if you're a big guy who's on the football team and jogs 10 miles each day whereas your father at the same age was a small guy who put in a solid 3 hours of chess practise each day, then you are going to need to eat a lot more than he did at the same age.
Prove that you're not just being greedy. Work out your BMI (body-mass-index) and demonstrate that it's not higher than it should be (assuming that it isn't - if it is then you maybe need to rethink this whole thing).
I hope that's all helpful, feel free to come back to me with questions or comments.
I suggest you simply show your parents this question and answer and ask to discuss it with them (remind them that you haven't embarrassed them in front of the world because no-one here has any idea who you are). I'm a parent (although my kids are a lot younger than you) and a lot of other people here are too, so hopefully some of us are seeing the parents-eye view here too.
PS - porridge is cheap, filling, healthy and easy to cook. Making yourself an extra bonus meal of porridge occasionally can help.
PPS - I've just noticed that you're vegetarian. OK, you're probably good on the "5 a day" front, but you are going to need fairly vast quantities of most vegetables in order to reach the likely calorific requirement. Consider snacking on things like nuts, which are pretty high-calorie.