I can see this from both sides of the fence!
As a parent with the way the current situation is overseas I can understand why your parents would not want you to join the army - they would be so worried!
Mother bit over!
I can also understand why you would want to do something that will bring more to your life! that said the army still gives you a normal mundane job to do 9-5 a lot of the time other than when you are on exercise, or overseas training. My hubby was in the army for 10 years and yes he did a bit of travelling but he also did much of the same things day in day out, thats why he left!
It can seem exciting and if it is truly what you want to do then dont let to much stand in your way. Asthma would not necessarily rule you out of the army they would just do a much more thorough medical on you. Also your Asthma might limit what roles you are able to do in the army, you might end up at a desk job and that isnt what you want.
Why not have a good talk with someone at your local army careers about the future opportunities for you. If you dont live near an army careers office you can go to
http://www.army.mod.uk/ this has a facility that will allow you to email an questions that you might have.
I hope that you can work this out with your parents but I do think that to go to them having looked at your options in great detail would show them how serious you are.