According to this webpage, under the payload shroud were the airlock, the multiple docking adapter, and the Apollo telescope mount. The latter was folded in front of them but swung to the side on orbit entry. The Saturn Instrumentation Unit (I.U.) was also above the third stage.
Dragon’s landing parachute is stowed roughly in the nose cone hinge area, and this leads me to believe the nose cone has to open for the chute to deploy.
That belief is incorrect. Other than both being installed near the top of the vehicle, the parachutes and the nose cone have nothing to do with each other.
In normal operations, I believe the nose cone is ...