This is what about.com says:
http://usmilitary.about.com/od/joini...y/a/asthma.htm
I don't know for sure, but my friend tried to join the US Navy, but got turned down due to his ADD. He later asked the US Army wether they would accept him and they said they said they would. They told him about the changes made on this matter.
As for your folks putting you down on that, take my advice:
My dad was fine with me joining the Canadian Army (Officer in the Armoured Corps most likely). My mother wasn't, she outright refused. For a while, I was really dissapointed. Then I discovered, this is my life, not my mothers. She can give me her advice, and I welcome it, but I'm the one who ultimately makes the desision. Keep that in mind, this is your life.
Hope this helps, and best of luck to you.
