Burnett's 1^st Texas Battalion was involved in the defense of Vicksburg MS in 1862-63. James Parker enlisted in Grayson County on Aug. 1, 1862.

Since James Parker took the oath of allegiance and enlisted in U.S. service after deserting in 1863, it's interesting that he later spent time in a Confederate home. I'm assuming that you have not seen his military service files. Copies are available here --