It was a P3D shadow setting...
I had the Internal Vehicle setting turned on, and when I turned it off, the results were much better. When I solved this problem, I also downloaded TomatoShade and the R&D preset. I also changed the HDR values as you can see in the picture above. Here is the new result:
That's already much better, but the buttons on the GP (FCU in Airbus terms or MCP in Boeing terms) are very hard to read because there is a lot of shimmering on the buttons. The overhead panes still is very dark, and because of that, the overhead panel is almost impossible to read. I think I should make a new topic about the shimmering.
Here is a picture of the very dark overhead panel:
@scoobflight Thanks for your help so far!