For interested parties
About 2008 I issued a request for 16g choke measurements and a number of respondents provided data for Parker, Fox, Baker, Ithaca, and Smith guns. Almost all had a choke length of 1 1/2 to 2 inches with a very short parallel section at the muzzle. 
16g Parker guns made from 1891 to 1927 with chokes from .002 to .040 all demonstrated a choke length of 1 1/2" to 2". 
Fox 16g doubles made in 28', 33', and 37' were found to have chokes about 3 1/2 inches long in the tighter choked barrels BUT a 36' 32-inch barrel A-grade had full chokes of only 2 1/2 inches.
I have no data for 20g
Dave Miles provided some excellent data from the PGCA showing that in full choke 12g. Parkers, the choke was bored with between .007-.010 constriction between 4 and 2 inches with the remainder of the constriction in the last 2 inches. 
This is similar to what was found in a limited sample of 12g. Lefever, Remington, and Ithaca guns. 
From Austin Hogan
Parker Chokes prior to sometime during Remington's ownership were cut with the adjustable nut borer and had the characteristic "OGEE" shape, with a parallel section 1/8 to 1/4 inch long. Late Remington barrels were reamed from the muzzle end, and have longer parallel sections. 
P. 394 of L.C. Smith "The Legend Lives" shows the 12g choke specifications; a 1 9/16" taper segment with a 1" parallel segment.
 
I'm currently shooting a 1906 Smith 0E 16g shortened from 30" to 29" without apparent POI issues.