Wonderful piece of family history. Too bad it was improperly refinished.
I can make myself believe I can see the Remington date code on the left barrel flat in your first barrel flat picture. Looks to me like it is KF K = May F = 1937 --
240287 02 Date Code annotated KF K = May F = 1937.jpg
Here are the Parker Skeet Guns from the Price List dated May 17, 1937 --
Parker Skeet Gun, May 17, 1937.jpg