What will you use as a propellant? Compressed air into the bottle?
In any case, the bottle should fly upside down, bottom up, probably stabilized by some attached tail, as a real rocket.
The best place for the egg would be on top of the bottom of the bottle, inside a sort of conic capsule with the parachute folded & attached to it.
The top capsule can be obtained cutting the upper part of a similar bottle. It should be kept in place, not attached, by means of 3 or 4 lightweight sticks attached to the "cone" and surrounding (not attached) the main bottle.
The capsule should separate when the bottle stops pushing and starts to fall (as a discarded booster). The tails of the rocket could help dragging the bottle, when the thrust is over, to separate it from the cone (which should have more inertia because of the egg inside).
When the capsule (heavy because of the egg) starts falling fast, the parachute should unfold from the rear of the cone because of the air drag.
The parachute shall be made with very lightweight plastic foil and attached to the cone containing the egg by thin cotton or silk threads.
The dimension of the plastic foil and the number of strings should be found experimentally by dropping the parachute from an altitude (75 feet?) with attached a weight to simulate that of the egg.
Have fun. I hope it will work.