I jumped out of a chopper and went splat because I didn't know what to do.
2 Answers
On the XBOX 360, open the parachute by pressing "a". You must be past a certain height threshold for your parachute to be deployable. A double press may be required to open your parachute.
Source: http://www.battlefield3gamer.com/apps/blog/show/9807786-how-to-open-parachute-in-battlefield-3
As long as you are in "freefall" mode (you can see both of your hands in front of you, and no weapon), you can press the jump button (default "A" for xbox) to deploy your parachute. You will enter freefall mode after falling for a certain amount of time/distance.