I think to compare Parker to Boss or Purdey is just foolish. Entirely different kinds of companies making different kind of product. Just as we would not compare Ferrari with Volvo.
The OP starts really with a different question.....are they the best of American factory brands.
What gets missed by many in these types of discussions and it may be easier for a non American to see it, is the nationalism at play. Collectability, current market pricing....these things are driven hugely by a desire for Americana, for lack of a better term. And the value due to that aspect affects the judgment about how Parker (and the rest of American guns) fit into the wider world of shotguns.
Take off the "American" or the "English" hat and judge any and all guns based on design and workmanship. I don't know where Parker would line up in that but I doubt it would be anywhere close to Boss or Purdey.
I'm Rick by the way, I totally get where you are coming from, not comparing Purdey, Boss in the same quality i.e. side by side as in next to each other, but looking at it regionally, you have American makers that run the gambit and then you have English,and so forth and within their regions they are tiered by quality, fit and finish. So the Q wasn't saying the Parker was on Par with Purdey, but giving a curve to American makers "would Parker have been the Purdey of America"? I hope that makes sense?