Retracting firing pins.

[B. Dudley]

Here are a few things to check...

With the back half apart from the barrels. Just like it is in your photos. Refering to your second photo. Inside the water table you will see the cocking cam. This is the silver piece down inside the groove. The cocking cam is what is lifted by the cocking hook and operates the cocking slide. Under that cam is the unhooking slide. With the front of the action facing you, push in on that pin that sticks out of the front of the hinge pin area (on left side when looking at it as in your second photo). This is the pin that needs to be pushed in by the forend when puttin git on. When pushing in on this pin you should be able to look down in and see just the tip of the unhooking slide move in and out under the cocking cam. That is all that will move. If that is working, then the unhooking slide is good.

With the barrels on and forend off, the barrels should go on and off freely with no obstruction.

With the barrels on and the Forend ON, start to open the breech. If the tips of the pins do not start to retract after about 1/4" of opening, then they are not cocking. Another note is that the extractor SHOULD be timed correctly so that it will not hit even the uncocked pins while opening. Both of my Parkers that I am looking at here will not hit. I actually have to pull the extractor forward with a thin screwdriver or something to get them to contact the pins when opening. And that is when I do not have the end on as to simulate a gun that is not cocking.
Another thing if it is not too much is to take the extractor right out of the barrels. This is done by removing just one screw on the bottom of the barrel locking lug, in front of the cocking hook. Then the extractor pulls out the breech end.
Removing the extractor will allow you to open the action fully with the forend on and have the extractor not hit the pins. When the breech is fully open this way, you can see the cam being lifted near the end of travel. That is IF the gun is cocking.

The final (or even first thing) to check is the actual cocking hook on the barrels. This should be sticking down perpendicular to the direction of the barrels. there is a spring behind it and it should have firm pressure when pushed towards the muzzle. And it should spring back. Your issue might be a sticking cocking hook or even a broken cocking hook spring. If the cocking hook is not in the perpendicular position, then it will not engage with the cocking cam and the gun will not cock.