Packing
Packing a parachute makes it as small as possible, and it's one of those things that needs to be done correctly in real life, lest the wearer's chute fail and they fall to the ground. The object of packing is making it as easy to open naturally as possible, so that the parachute simply sort of rolls open and expands to full-size. On real parachutes, this is sometimes aided by a small leader chute that pulls the main chute out. Of course, on toy parachutes, the stakes are far lower--your toy soldier is unlikely to die from a ground impact--but in order to get the results you want, you'll have to fold the parachute properly to start with.
Folding Flat
Begin by laying the parachute out flat on your work surface. Grip the parachute by the center point on the material, and pull the corners of the chute together by the strings--allowing the toy or whatever is attached to the bottom to dangle.. Make sure the chute lies as flat as possible. Lift the corners up and make sure everything lies flat, and that the fabric is not trapped by other parts of the parachute. If you are a fan of origami, this will look a little bit like the Water Bomb Base when complete--you can see this fold under Resources.
Rolling the String
Fold the top point of the chute down to where the strings meet the fabric, loosely. Do this a second time to create the packed parachute. Then, wind the remaining string and parachutist around the packet--again, loosely. The parachutist is now ready for a test run. Be prepared to find that your parachute is still wound too tightly or too loosely: if this is the case and the chute fails to deploy, it will be obvious. If you're concerned about the toy being damaged in this experiment, you should try with a more sturdy toy first, until you have a good feel for how to fold the parachute best.