Dudeman posts about the chutes at DZ -
dudeman17
#61842
A pilot chute and a drogue are similar in design and appearance. The difference is in the function. The drogue trails behind something/someone in freefall for the reasons stated above. The pilot chute deploys the canopy. In the 'cutaway' method of emergency procedure, then the reserve would have a pilot chute, and that's what is seen in that last picture you posted.
Again, I would guess that any white material might have come from the dummy reserve.
The pictures in the 'citizen sleuths' link:
In the picture of the canopy laid out on the table, the lines at the near end, the far end is where the pilot chute would be, yet nothing is seen. No pilot chute on that rig.
That bundle of lines between the hardware in the container is a cross connector. The parts of the hardware that can be seen inside the container, those rectangular metal loops, are where the parachute lines attach. That hardware goes through holes in the container fabric, and the other end are those hooks/clips that are seen in other pictures. Those clips attach to the D-rings on the main harness. If one of those attachment points were to fail on opening, you'd lose tension on half of the lines, the canopy would collapse and you'd go in. That cross connector would hold that side of the hardware and prevent that collapse from happening.
Those lines tied to the side of the container is probably something Cooper did in an attempt to tie that to him. That those lines are white, and not pink like the ones on the chute in the picture make me think that they too might be from the dummy reserve.