This is a real possibility as this happened with several other men who originally joined Dawson's 19th Arkansas under the Conscription Act. These men were later reorganized and some of those men were transferred to the Portlock's 24th Arkansas, and later the survivors that were not captured at Arkansas Post became Hardy's Regiment Ark. Inf. which was later assigned to Frost Missouri Brigade and many records found their way to Missouri Archives.
Do you know if John Parker had any Brothers that may have joined with him and where did they enlist from? Maybe Calhoun County?