Chris:
That is a sweet looking set, as good as it gets. I am glad someone that cares about it has it in their hands now. An amazing place to start with Parker Bros. shotguns.
The reason there are no records for your gun in the PGCA archive is: books are missing. They are not necessarily destroyed or lost. They were just not available for the team that made the copies from the archives of Remington. The speculations about the history and possible location(s) of the missing books are found throughout this board.
Read all about that in various places of this board. Here is a starting place for you:
https://parkerguns.org/forums/showth...ighlight=Books
Here is one (1) of the books about Parker guns you will be interested in buying for yourself, by Price & Fjestad: Parker Gun Identification & Serialization. Even if your gun is not listed, the next gun you look at may be in the book, giving you a solid reference for the original configuration.
https://www.amazon.com/Parker-Identi.../dp/1886768374
If you care to really learn and want to study these guns, just go ahead and get the book as soon as you can. If you don’t get it now, you’ll get it later and wish you had it before.
Also, this looks to be the first time this set has come to the attention of the PGCA discussion board, in the current version. That is based on using the search function.
In addition, the records search at Griffin & Howe did not bring up this serial number.
http://secure.griffinhowe.com/Research-getserial.cfm
And even if it had come up, getting to them is a challenge you can read about in postings on this board.