For lines, perhaps just single polygons would work - like when rigging you'd stick them between the chute and pack - so you'd have 2 verts stuck onto the chute, and the other one on the pack - them when animating you just forget your lines because they're already taken care off - they're just unrealistically stretchy - but you'd never notice, you'd just see these nice thin wires on your parachute when it's opened up.
Doing what BigAdd did is a good idea - like make your parachute all opened up, then rig it and fold it up with the animation, playing that in reverse would be your basic opening animation.
Van-B
Put those fiery biscuits away!