I have looked at loads of videos on how to remove the rear door panel (2006 Marquis). None show a 2006 which does NOT have a "pull-bar" held in place with two screws. I see only two screws (One is actually a Torx screw in the door latch cavity, and the other is down on the lower left of the panel) There are no covers visible.
There is the shiny Plastic (fake wood trim) that I hesitate to pry away and break, or the "armrest" that might hold the answer???

 

Remove all of the screws that are visible.
Pull straight up sharply from the bottom edge of the panel.
It should come right off.
If not, you may have missed a screw.

 

 

I have an '04 Grand Marquis GS which I think is very similar to yours.
First: DO NOT REMOVE THE WOOD PANEL OR THE ARM REST - THEY ARE NOT REMOVEABLE WITHOUT DESTROYING THE PANEL. IT WOULD THEN NEED TO BE COMPLETELY REPLACED.
1. Remove the trim screws you see - one at the bottom of the panel (7mm).
2. Remove the little cover over the torx in the door handle. The door handle itself may be left snapped into the panel until re-install.
3. GENTLY pry up the door window switch panel (bezel) as the clips are prone to break off and a replacement bezel costs way more than you would think looking at it. Carefully unclip the wiring harness from the bottom of the window switch.
4. Under the space now open you will see a hex head bolt (12-13mm) which is attached to the door with a speed nut. This is what does the most holding the panel to the door. Remove the bolt.
5. Lift the the panel to remove from the door.

Installation is the reverse.

I realize that my reply is 2 years late, but perhaps it can help someone else. I have been into all the doors of my 04 to replace all the window regulators, speakers, replace a mirror, hunt down door rattles, adjust windows, etc. All the door panels are pretty similar in construction, with the drivers door just having more wiring harnesses to disconnect. BE CAREFULL OF THE WINDOW/LOCK BEZELS - THEY ARE GETTING SCARCE AND EXPENSIVE!