Twist barrels on lower grade guns may have been thicker on some guns----I just don't know for sure. However, Parker Bros. made the PH grade gun on three frame sizes, 1 1 1/2 and 2 along with a few #3 frame guns with twist barrels, and they also made several higher grade guns on the same frame sizes with fluid steel and/or higher grades of Damascus barrels. Since the widths at the breech for a given frame size is fairly constant with Parker guns, the thickness of the barrrels ( in the chamber area where it matters) are about the same on a gun of a given frame size. Until the VH grade was introduced, the PH was the lowest grade Parker gun and most of the time that it was offered, it was available only with twist barrels.
I really doubt if there was or is a huge difference in strength of the twist vs. Damascus but I believe that the Twist barrel is inherently a little weaker because of the gemetry of the way it is fabricated. Any hoop stress at all will tend to pull the welds between individual strips apart in the longitudional direction in the twist barrel whereas the twisting of the bar in Damascus barrels precludes this posssibility. The three-iron barrel on the GH Parker that I donated to Tom Armbrust's destructive test withstood over 30 KPSI before bursting. I really doubt that a twist barrel on a #2 frame PH gun would have done as well.
I have no doubt that the cost and desirability of the fancier Damascus barrels was the difficulty in making them and the pattern rather than the strength, but a slight increase in strength was also a factor IMHO.
With regard to Greener's concern of overtwisting with the finer patterns, I don,t see any evidence of this on barrels that I have refinished, but his concerns may have some validity.
