Well, as it turned out I had to remove the sail panels, and some other parts. Here are the steps I followed, and I did manage to get the parcel tray cover out without damage.
1. Disconnect the battery.
2. Lift the rear seat back off its retaining hooks, and rotate it forward away from the rear tray.
The seat back hangs on these hooks.
3. Remove the two metal filler pieces between the seat back and the rear panels.
Metal filler pieces, one on each side.
4. Remove the interior trim around the back window. There are two large aluminum L-shaped pieces, and two smaller black painted L-shaped pieces at the bottom corners. The center black piece at the rear of the window stays in place.
Metal window trim pieces.
This one stays in place.
5. Remove the two courtesy light assemblies from the sail panels.
Courtesy light assemblies (1965 Impalas didn't have a dome light).
6. Remove the left and right sail panels. Each sail panel slips under the aluminum window trim above the door windows and quarter windows, and each panel is held to the sheet metal with one push-in type trim clip.
Side panel after sail panel is removed.
This is the trim clip.
7. Carefully pry off the paper and thin foam rubber that are glued to the front of the shelf, behind the rear seat.
This part is glued to the metal behind the seat.
8. Carefully raise the center of the parcel tray cover, and pull it away from the side trim panels.
9. Carefully ease the rear of the parcel tray cover out from under the remaining (black) center section of interior back window trim.