Bryan, I agree with you, now that I know that there was a W. H. Parker in the Randolph County Census of 1860. I checked the Adams Regiment roster on the Gerdes web site and there are no William Parkers of Carroll County that I can find.
I've also been through the William Parkers listed in Walls Index of Confederate Soldiers in Arkansas and have pretty much eliminated all of the Willima H. Parkers and William Parkers. In some cases, when I couldn't tell from the information in the Gerdes transcriptions, I ordered the man's papers from the Arkansas History Commission.
Sooooooooooo, where do I turn to dig next? My gt. gt. grandfather was 32 in 1861 and that is a "hard" date. Most of these guys I've been seeing can be eliminated on age alone. Of course, it is entirely possible that he was not in the War, although family tradition says that he was.