I'm like Francis. I will tear down my own stuff and have now played with three rough Parkers, with no end of assistance from the gentlemen on this forum. Galen, you say the hammers won't drop but the triggers will move. Does the rear trigger (left barrel) move far enough to be flush along its full length with the trigger plate? If it does, the hammer should be released. If the hammer is not released, there may be a broken sear spring or broken sear or an issue with the cocking mechanism. (I chose rear trigger because it is much easier to verify extent of movement, i.e., flush to plate than it is with the right trigger) If the trigger only moves a bit but not flush to the plate, then something is interfering with sear release, hammer travel, mainspring travel, mainspring piston cup travel. If the gun has never been opened up, there may be a good likelihood of internal rust seizing the mainspring piston cup, thereby not allowing the mainspring to release its tension. Even when barrels and external metal have been maintained, after a century of time, the internal action components can get into rough operating condition if the action has never seen any servicing, especially if it sits in disuse for decades.
Since you have a nice gun and if you feel hesitant to play, take it to a recommended Parker gunsmith and then enjoy it for years to come. The Parker is a fun shotgun to shoot, especially in the field.
