It sounds like your parents are the "no nonsense, let me see the facts" kind of people, so here's how you should break it down.
1. Look into the arguments Peter Singer uses, he's really good.
2. Your Dad's little "veggie farming is killing the ozone layer" speal is bullocks. Think about it in lamens terms, If humans eat the animals, the animals will have to eat something too, that something will most commonly beeeee THAT'S RIGHT! VEGETABLES!! So! by us eating the vegetables, we are skipping a step, which eventually, will result in a supply and demand domino effect, which will mean that less and less animals will be being "produced" for human consumption, meaning less Co2 in the air from the smelly farts and slaughter houses etc etc etc. So rather than having an area for veggies and an area for animals, we would just need the area for the veggies
3. Work out the price of your current meat an three veg meal, then work out the price of a vegetarian meal. Show them how much they would save if you went vegetarian.
4. Print off an info sheet about exactly how much iron a person needs per day and which vegetables etc you can get it from.
If none of these tactics work, tell them you are going vego whether they like it or not, and if their caveman views are going to stop them from having a good relationship with their child, then that's their own fault and you shant be eating with them again. Then make the most delicious smelling vegetarian meals until one of them caves and wants to try it as well.
I hope these help, I realise not everyone is as strong headed and bossy as me, but this way you can present solid facts to them (although this doesn't always work, I was teased to no end by an old group of friends and they didn't hold back until one day when I collapsed in tears and they realised how much being vego meant to me).
Good luck!
If you ever go vegan, I highly recomend
www.parsleysoup.co.uk she's so inventive and delicious!