Your focus should be less on the parachute and more on what container the egg will be in. In my experience, students who spend their time creating a good container do better than those that spend their time creating a good parachute. It is possible for an egg to survive a free fall with no parachute as long as it is in a good container. In order for the egg to survive, you want to minimize the force that the egg experiences when it hits the ground.
According to Newton's 2nd law, Force = mass x acceleration. Since the mass of the egg will be constant, in order to minimize the force, you need to minimize its acceleration. When you think of acceleration in its true scientific form, which is the rate of change of velocity (acceleration = change in velocity / change in time), it begins to make more sense. You want to increase the amount of time the egg has to stop as the container reaches the ground. This is exactly the same reason air bags in cars are successful; they increase the amount of time your body has to come to a stop, thus decreasing the acceleration and the amount of force your body experiences.
In order to accomplish this with your egg, think about air bags. They inflate to large sizes to the point that there is no space between the back of the seat, the person, and the air bag. In the same way, you want your egg to not be able to freely move in the container. The air bag provides a way for your body to continue moving safely even when the car has stopped. The same goes for your egg. Think about how you can provide it a way to continue moving even once the container has stopped. This will provide more time for it to slow to a stop, which will reduce the acceleration and thus the force the egg experiences.
Once you have a good container, go ahead a build a parachute for good measure. Good luck and have fun!