Sally,
There are 5 James Parker's from Texas listed at American Civil War Research Database. One was in the 1st Tex Sharpshooter's. Also there is an S.A. Willis in the 1st.
There is no history for this unit there.
I can't find my notes right now but beleive they were mustered in, in Aug 1862 became part of Maxey's Brigade,and were in Mississippi untill sometiime in late 1863 or early 1864. In Feb of 1864 thet were assigned to the Indian Territory in Sep 1864 they were assigned to Cooper's Cav Division. Mustered out May 26,1865
Mike