Hello. I respectfully suggest that the four guns identified as "3-inch Parrott" in the Federal reports of Fort Hidman's capture, by Capt. Jenney (filed under Sherman, and including the sketch that, with some minor changes, eventually ended into Battles & Leaders, v.III, p.453 [ar24_760]) and McClernand (who likely copied from his subordinate [ar24_705]) are classic 2.91-inch Parrott instead.
My reasoning is based upon:
Source 1) James C. Hazlett, Edwin Olmstead & M. Hume Parks, Field Artillery Weapons of the Civil War, 1988 revised edition.
Source 2) Edwin Olmstead, Wayne E. Stark & Spencer C. Tucker, The Big Guns: Siege, Seacoast and Naval Cannon, 1997 (title notwithstanding, this book's appendixes list anything larger than a musket, updating and correcting the first book above).
Reasoning A) 3-inch Parrotts were rare indeed in January 1863. No Federal one was accepted until February 12, 1864 (source 1, text, p.113, & appendix 4C, p.233 & Source 2, appendix D16, p.271). The record of Confederate ones (apparently all made by Tredegar) is scanty. According to Source 1, there are just four surviving pieces (text, p.113) corrected by Source 2 in seven pieces (appendix D17, p.273). Only one can be dated (cast 25 November 1862, now at Manassas NBP). Two others (now at West Point, NY) were captured at Port Hudson, and so cannot have been cast much later. But keep in mind that most or all of these pieces may have been rebored at a later date for uniformity of caliber, as the Federal did in 1864-65 with nearly half of their earlier production (Source 1, text, p.113).
However, we agreed (or so I believe) the Fort Hindman guns originated from Pike's domain. So, they would have been cast not later than early spring 1862 (but probably before). They could have either originated in Richmond (i.e. Tredegar, from 18 November 1861, Source 1, text, p.111) as per Capt. Woodruff's memoirs (With the Light Guns in '61-'65, p.64) or in New Orleans (i.e. Bujac & Bennett, from 16 December 1861, Source 1, text, p.113) as per Pike's remarks to Jefferson Davis, 31 July 1862 [ar19_861]. But the latter point is open to interpretation, and he may have been referring to bronze guns. Also, some Parrott were cast in Vicksburg and Memphis early in the war (Source 1, text, p.113). In all these cases, the pieces were definitely 2.91-inch ones.
Reasoning B) To the uninitiated (me?) 2.91-inch and 3-inch guns of same exterior appearance are virtually identical. Actually they may looks the same also to trained artillerists! This is the Gettysburg report from the 2nd Corps ANVa Chief of Ordnance:
There is complaint made by Lieutenant, [William] Fontaine, ordnance officer, Jones' battalion, that the ammunition of the 3-inch (banded) gun, or navy Parrott, is mixed up with the 2.9-inch 10-pounder Parrott in such a way as to cause great inconvenience. Two guns were rendered unserviceable after firing 12 rounds, from the shell lodging in the bore. [ar44_458]
Lieutenant [N.M.] Osborne, ordnance officer, Carter's battalion, reports also that some of the 3-inch Parrott ammunition was issued to him for the 2.9-inch Parrott ammunition. Lieutenant [John] Selden, jr., ordnance officer First Virginia Artillery, also reports that he received some of the 3-inch Parrott ammunition. [ar44_459]