J. M. Cox, age 23, Private, Captain A. F. Moss' Company, 4th Regiment Texas Dragoons,* joined for duty and enrolled October 28, 1861 at Camp Hebert, near Hempstead, Austin County, Texas, by Capt. A. T. Moss for 12 months, no other records**

* This company subsequently became Company K, 12th Regiment, Texas Cavalry. Also known as the 4th and as Parsons' Regiment Texas Cavalry

** The next surviving muster roll for Company K was that for the period September 1, 1863 to December 31, 1863. Missing are the 6 reports for the intervening period. As he was not recorded for this last muster, he would have had to been dropped from the rolls sometime before August, 1863, for discharge, death, capture, desertion, etc. This leaves his fate uncertain.

Here are a few sources which may be helpful in your search.

12th Texas Cavalry Regiment, "Campaigning with Parson's Texas Cavalry, C.S.A. The War Journal and Letters of the Four Orr Brothers, Twelfth Texas Cavalry Regiment,", edited by John Q. Anderson. Published in 1967 at Hillsboro, Texas, a copy costs around $ 75.00.

12th Texas Cavalry Regiment, "Between the Enemy and Texas: Parson's Texas Cavalry in the Civil War," by Anne J. Bailey. Originally published in 1989 by the Texas Christian University Press, Fort Worth, Texas, 355 pages, maps, roster, cost $ 40.00. Parson's Texas Cavalry Brigade in the Civil War was comprised of the 12th Texas, the 19th Texas, the 21st Texas and Morgan's 13th Texas Cavalry Regiments and served in the Western Theater fighting in Arkansas, Missouri, and Louisiana. The book includes detailed quotes from soldiers' letters.
http://www.mosocco.com/texas.html

These two gents are considered "well versed" in the regiments history:

12th Texas Cavalry 1861-5 jsanders@jaguar1.usouthal.edu
12th Texas Cav, Co A 1861-4 wier@calcite.rocky.edu (Bob Wier)

You might also try a Goggle search 12th Texas Cavalry