Boy do I even you- We tried twice here in MI to get a legal dove season, and the antis-aunties-and Audubonistas shot us down in flames. I get second best- no season and no limits, barn pigeons, and see some doves on occasion when after the "Sky Carp" on area dairy farms.
If you consider pigeons and mourning doves to be "cousins" I can tell you this, from almost 60 years of shooting them-- birds in flight can maneuver like a F-16 fighter, but when flushed from a bare branched tree or a power line, both dove and pigeons will drop down to load their wings before they begin a power climb. I have never shot snipe or railbirds, but would imagine a walk-up hunt in a sunflower patch for dove would give you some real zig-zag maneuvers to test your gunnning abilities.
I remember reading in Bob Brister's fine shotgunning book about a Texas dove event, a social event that turned into a big money bet between a young 100 straight skeet shooter with a 28 gauge and an old man (I'm in that category now) and his old well worn 12 bore side-by-side (make and grade unspecified)-- Bob Brister also mentioned the need for tan and air vented camo type clothing, and cool drinking water, not just for the shooters but also for the dogs, and staying in the shade- and to bring plenty of shells.
I'm 99% a 12 bore man, so I'd take either my Smith Grade 2 or my Parker PH- both have 30" barrels, DT, extractors (I reload) and are choked Mod. and Imp. Mod or Full- BUT- at my age, I don't get on the birds as fast as younger men might, so the tighter chokes and longer barrels suit my pass shooting style best- I use RST 1 oz. No. 8's or Rem light RST green 1 &1/8 oz. also no. 8's
It is fine that you have two Parkers, a 20 bore and a 28 from which to choose-I only own 12 bore "Old Reliables" so the choice is easy in my case.
I have read and re-read Nash B's "The Dove" more than any other of his "How To" stories- and have benefited from his sage observation that: for a right handed shooter (me) when you swing a shotgun from right to left, the muzzles climb slightly, but from left to right they tend to drop a tad- Words of wisdom--IMO