The evaluation of originality provided by DAM16SXS was probably not the news that the OP wished to hear, but his observations seem quite accurate. Unfortunately for Parker collectors, there is probably no other American shotgun that has been so subject to refinishing, upgrading, restoration, and even outright counterfeiting. Parkers became desirable and collectible, even in the lower grades, so there was a natural tendency for people to try to enhance the value. And they have been collectible for so long that decades old restorations have aged and now sometimes don't look too good to be true.

The comment about serial number stamps in the wood, and metal, is quite valid. Matching serial numbers are always desirable to the collector, and it is no great trick to stamp new numbers, or even engrave them on higher grade guns. But many newer sets of number stamps utilize different styles or fonts than what was originally used by the factory, so it pays to study small details like that.

Evaluating original condition also requires one to look at the sum total of every part of the gun, and carefully examine whether normal wear really appears normal over the entire gun. This particular Parker is 132 years old, so it isn't surprising to see that the case hardening colors on the frame are quite faded, and that the top lever is very slightly left of center. That indicates that this Parker has seen use, and is not some pristine virtually unused closet queen that is exceptionally well preserved. Yet the wood finish and the barrel blue do not exhibit that same level of normal wear. Even when wood finish is not worn, it changes appearance as it oxidizes over such a long period of time, and perfectly filled pores on an old gun can provide an important clue. The checkering also appears way too clean, so that combined with the other details DAM16SXS mentioned, such as the subtle shape of the fluting at the nose of the comb and the special order checkering pattern even more strongly suggest that the stock is likely a replacement. Assuming that is correct, whoever did the work did a very nice job.