I think the important thing here is to consider where exactly your parents' concerns lie, and then do your best to alleviate those concerns. As was said earlier, it is a possibility that they may never agree with your choice; only you can tell how difficult that would be to deal with in your own particular case. I think I can understand parents being concerned regarding the danger to life and limb. Naval aviation has never been the safest career, and probably some parents will tend to envision their son or daughter being killed in a horrific wreck or disappearing somewhere over the Pacific, that sort of thing. But once you're an adult, you are the one to choose that risk. I for one want to serve my country, and while I have no desire to die in combat, I am willing to join the many thousands who have given their lives if it is required of me. Freedom is not free. Your parents and grandparents sacrificed for what you have, and it is your responsibility as a citizen to do the same. Moving on, I suspect that matters of economics and such are less likely to be a primary concern, but I think it should be fairly easy to demonstrate that you're not going to starve. Another thing is that a fair number of parents don't want to give up their vision for what they think their children should do with their lives, and again, at some point you're the one who ultimately needs to make the decision. Finally, don't think you need to make the decision without your parents input. I guarantee there will be a lot more hurt feelings if you sign up against your parents wishes without at least sitting down and telling them what your plans are, what the benefits and risks are, and giving them the opportunity to let you know what they think. You might learn something, and I doubt you'd regret it. If they still don't agree with your wishes, it is then your prerogative to decide whether or not they have good reason for not wanting you to join and if you are going to go against their wishes.
My sister joined the Navy last year to go into linguistics, and my Mom especially was against that. So far she hasn't expressly changed her mind on it, but my sister seems to be doing well and my Mom doesn't seem worried about it any more. In my case, my parents have been hesitant about my wanting to join. Part of it is that they think it might be better for me to go to a flight school and get my commercial and Airline Transport License, and also I think they're of the opinion that Navy aviation is a jock's world that I would have a hard time fitting in to. Basically, on the first one I don't think they realize how tight the airline industry is right now and how strong the likelihood of me being stuck with a pile of debt and no flying job; on the second, no, I'm not at all a jock, but I've been around enough and am athletic enough to get along with them, and I think they overestimate on that count. The other major concern that comes to mind is political. Both my parents and I have no problem with the idea of me serving under the current administration, although I am not about to suggest that it is without it's issues. The concern is what things might change if the likes of Hillary Clinton or John Kerry were to become president. I don't want to get into a political discussion here, so I'll just say that the philosophy regarding the military was vastly different under Clinton than it is under Bush, and I think my parents do have legitimate concerns about what a change in leadership would mean. All in all, I consider it worth all the potential risks, and if I am offered a flight school slot I can hardly imagine turning it down.