Carl, I feel sure that William H. Parker served in some capacity in the Civil War, either in the Confederate or Union army, in an Arkansas (or possibly Missouri) regiment. Though lots of men wanted to remain neutral -- for personal, religious or political reasons -- neutrality just wasn't an option. The Confederate Conscript Act was rigorously enforced (in those parts of Arkansas which remained under Confederate control), and those living in areas subject to Union control were subjected to great pressure to join the Union army. Threats to the man and/or his family were commonplace.
The problem is that so many Arkansas soldiers have no documented Confederate service, due to the fragmentary nature of the records. There is virtually no record of men who enlisted after the winter of 1863/1864, unless they were lucky enough to have a supplementary record as a prisoner of war or as a pensioner. Sometimes you get lucky and find an ancestor mentioned in a letter, diary or journal.
I suggested Adams' Arkansas Infantry as a possibility for William H. Parker, simply because of its Carroll County connections and the fact that most of its records were destroyed. That would account for his not being listed in any indices. Of course, finding the "smoking gun" is probably going to be a matter of luck, more than anything else. I've been able to just about double the number of men identified as members of Adams' regiment by laborously picking my way through pension records, letters and a couple of very helpful postwar reminiscences -- but that still leaves about half the regiment unaccounted for.
If you want to explore the possibility of Adams' regiment, you won't find a pretty picture, though. Much of the regiment was composed of very reluctant conscripts. After a couple of near mutinies in the weeks leading up to the battle, most of the regiment deserted the field at Prairie Grove. Less than 200 men ralled to the colors and stayed in the fight. The regiment was disbanded in disgrace after the battle -- the only Arkansas regiment to desert in the face of the enemy.
I don't believe, however, that we in the 21st century should condemn those men. Most of them were in the lose-lose situation of not wanting to take up arms against Old Glory, yet not wanting to fight their friends and neighbors on the Southern side. But, as I said before, the men were not permitted the luxury of neutrality. They were conscripted and forced to serve against their will. Thank God none of us will ever be forced to make a choice like that.