Carl, I think it's highly unlikely that your William H. Parker is the W. H. Parker of Co. E, 38th Arkansas Infantry. This man is more likely the William H. Parker, age 20, in the 1860 Randolph County census. All of the original enlistees appear to have been Randolph County men, along with a few from bordering counties in Arkansas and Missouri.
Instead, you might want to explore the possibility that your William H. Parker served in Adams' Arkansas Infantry (also known as the 3rd Regiment, Northwest Division), which was organized in Northwest Arkansas in August 1862, and which was comprised of four companies from Carroll County, three from Newton County, three from Madison County, one from Pope County, and one undetermined. Only a couple of these companies have complete rosters; the remainder are highly fragmentary, so it will take some detective work. The regiment was disbanded shortly after Prairie Grove.