Sounds like you're on the right track to finding a balanced, healthy lifestyle that you can maintain.
A person can be "vegetarian" and still be completely unhealthy because they still choose junk food over wholesome foods. (I have a niece who is an example of this...she's been eating vegetarian for a couple years now, and has actually gained weight because she still snacks frequently on fried and packaged junk)
The key is whole foods...veggies, fruits, whole grains. If done correctly, plenty of protein can be taken in that way. But I'm with you in preferring some meat. So long as it's moderate portions, there's nothing wrong with eating meat of all sorts. (Again, we're talking fresh meats and not the nitrate-filled packaged junk) So yeah...whatever name you want to give that lifestyle...I think it's a great one to strive toward!
It sounds like your parents are doing some version of the Atkins diet. I tried that one. It does work if you're super careful and follow the plan strictly. And it can lead to some very painful side effects. I'll never do that one again! Plus, the moment you stop doing it, all the weight comes back faster than ever.
There's a sticky at the top of this forum with some great recipes. Also, another AWESOME resource is a site called SparkPeople. It's totally free...and has just about every tool and resource you can imagine for supporting weight loss efforts.
Good luck! And feel free to use the Journey Forum for blogging, venting, questions or whatever you need to help along the way!