Thanks, Bryan. I haven't seen Massey's work, but I have seen a similar one by Rena Marie Knight with essentially the same information. I obtained the service records for both of these W. H. Parkers. The man in the 1st Arkansas Mounted Rifles was listed as killed at Wilson's Creek and his age was given as 21, which is definitely not a match to the man in Gaither Cemetary, who is my great, great grandfather.
The man in the 38th Arkansas is the only one that was a logical option, especially since his service record had no age information or place of origin. Now that I know there was a W. H. Parker, age 20, in the 1860 Randolph County Census, I must concur with you. The man in Company E of the 38th Arkansas was not my ancestor. Oh well, the search goes on.