Just got my first Parker 12 gauge.

[Pa SxS]

We would need better pictures of the entire stock. Most of these early guns were straight grip and I notice your pistol grip trigger guard tang has 2 screws. I'm not quite sure if this is correct for a pistol grip although pistol grips weren't that common on those early guns. Give us more pictures of the gun to help us.

The best way to get rid of minor rust is use a bronze scrub pad found in hardware stores and gun oil. Go lightly and see the rust disappear. You can also use the bronze pad and oil to clean out the inside of the barrels.