I know you are just like I am. That you have read so much of this stuff over 30 to 40 years that you remember it but if someone wants documentation, chapter and verse, that is a difficult thing to refind because it is something that you have already done in accomplishing your own purposes.
As far as I can tell none of these Parrot guns turn up in Hindman command at Prairie Grove in December '62. Does that mean that they were in Churchill's command? I don't know. But the evidence seem to point a person in that direction. Hart's Battery seem to be a good possibility seeing as troops were dispatched to Fort McCulloch from White Sulphur Springs to get those guns. It would seem that some of these guns (possibly because of the lack of Harress and Limbers) were assigned to Fort Hindman itself. The fact that the diagram of Fort Hindman notes a 3" Parrot gun in the Fort would seem to be that it was more uncommon that the 2.9" Parrot gun which was more common of the two types of Parrot available, and being early in the war the 3" gun must have been a rare gun.