Straight grip, 10-gauge, Twist barrels should be a Quality S --
Price List, July 1, 1882, pg 20.jpeg
According to the dating by serial number on the home page, 37865 would be an 1883 vintage gun. The above Price List is dated July 1, 1882.
By 1900, Parker Bros. reduced the twelve entry-level Qualities to four -- Qualities R, S, T & U --
1900 Catalog Hammer Guns 02.jpg
eliminating the Laminated steel Qualities I, K, L & M and the Fine English Twist Qualities N, O, P & Q. A while later they they dropped the price differential for grip style and had only the Quality R & T. Still later they dropped the price differential for 10-gauge and had only the Quality T in the catalogs. But we see some late Twist barrel hammer guns stamped P, Aligning with the Twist barrel hammerless gun Quality PH.