Yep, my parents and some family flipped out a little bit at first too. My dad asked if I was crazy (CRAZY?!). My mom thought I was saying that to piss her off. Both of them were in the army around my age or younger, and this was back when there was no war going on and the army kind of sucked, so you can imagine what they thought it was going to be like with a war going on. So I kind of understand that.
My dad is more reasonable and settled down almost immediately after he thought about it a bit and decided he would be proud of me. My mom has tried every trick in the book to put doubt in my mind about going, including finding people who say that almost everyone washes out of flight school, that there isn't a GI bill for me (I'm going OCS), ad nauseum. I think eventually she realized that she can't stop me if it's something I really want to do, so she doesn't bug me too much these days. It helps if you've got other family or friends or anyone you know mutually who supports you for good reasons.
In the end, it comes down to living your own life. Your parents may be your parents, but they don't own you, and sooner or later they'll have to admit that to themselves as well. From my personal experience and in hearing the stories of others, it seems to me that most of the fury is a gambit designed to scare you away from joining in the first place. After you do, they might as well be proud of you.