Drew,

I asked the same question several years ago when I saw my first AAHE with Pigeon Gun engraved on its rib. I believe it is a model name used in their advertising.

I know that some British guns are recognized as "Pigeon guns" and a characteristic of them is the lack of a safety. But, I believe that Parker's "Pigeon gun" was a marketing term and not a type of gun. The rib of early AAH(E)'s were engraved Pigeon but all else is the same as what you would expect on any Parker (safety included). I have looked at early Parker paper and the AAH was listed as a Pigeon Gun but it didn't call out a specific configuration. At that time, the AAH would have been the top grade Parker available.

I suspect that advertising term was dropped around 1907, when the A1S was introduced.