Unfortunately, THOMAS PARKER was a fairly common name prior to the war. The Roster of Confederate Soldiers, 1861-1865 named THOMAS PARKER from South Carolina shows:
Parker, Thomas SC 11th Inf. Co.C
Parker, Thomas SC 21st Inf. Co.I
Parker, Thomas SC Inf. 1st (Charleston) Bn. Co.E
Parker, Thomas SC Mil. 18th Regt. Co.F
Parker, Thomas J. SC 14th Inf. Co.C
Parker, Thomas J. SC 1st Arty. Co.C
Parker, Thomas R. SC Inf. 9th Bn. Co.D
Perhaps someone else on this board can help you with the county break down and reporting points to help narrow this search down. In the mean time, contact:
South Carolina Dept of Archives and History
Columbia, SC 29211
Gen Info - 803-734-8577
Tell them who you are looking for and what information you do have and see if they can help.
I don't know if the county seat for Horry County would have records or not, but you might try them also.