abstellen is "to cut off / shut down."
That's one meaning of the word, what we need here is "putting down", "parking" or "putting in storage".
I don't know what Abstellpallung means. I can't find Pallung in any German dictionary.
Me neither. It seems to be a quite rare nautical term, meaning "strutting", "bracing" or "propping"1. A "Palle" is a (wooden) support block on which ships would rest in a dry dock.
The closest match I found in an actual dictionary is the adjective "pall" from the Low German language   which means "steif" or "fest" (stiff/firm) in German, and certainly could also be used as a verb ("make firm", to strut). Even the Duden knows the noun "Pall" from the same word origin, as nautical term for a pawl (ratchet) that fastens a capstan.
What's the black painted rectangle on the side of the fuselage?
It appears to be the point where the fuselage is meant to rest on a support frame for storage. I would guess that location is reinforced by the internal structure2, like @PeterKämpf said, for jacking it up.
1: Not sure what the subtle differences between these terms are, I'm not a native English speaker.
2: Or not.