Well, it's always a bit touchy when it comes to good family advice. I had similar thoughts before I joined up too. There are a few things to think about when it comes to this. The first is obviously, how freaked out they are. I mean, my mom was definitely not too keen on me joining up, but I knew she would still love me and that I wouldn't be giving her any really bad mental, physical, or health related problems. I mean, it sounds harsh, but how long are you going to live how your parents want you to? When do you get to start living your life the way you want to? These are all tough questions, depending on your family life. It's understandable to think this way, because some people tend to feel obligated to head in the general direction their parents want them to because, say for example, they paid for the education. It sounds like you have a pretty healthy relationship with your parents, so I would sit them down and really hash it out with them.
Then again, a my friends older brother joined the marines, buzzed his head, showed up for dinner, and THEN told his parents, knowing they'd have never have let him in the first place. The flipped, for about a week and now they are very proud of him. In the end, if they don't seem as though they're going to have a stroke or a heart attack (or both), and don't threaten to disown you, most parents end up being proud of the decision.
Most of the time, with parents, trying to get them to overcome their fears of what might happen to you is impossible. Parents always worry about their children. The best way to help them with it is to prove to them that you know what your getting into and tell them why you want to do it. Most parents eventually understand.