There very little in his record file. I tried access to the Official Records, but found nothing related to his death.

Robert C. Parks, Captain, (1st) Company B, Watie’s Regiment Cherokee Mounted Volunteers, enlisted July 12, 1861 at Prairie Springs, Cherokee Nation for one year, Lt. Colonel tendered resignation March 18, 1863, absent on detached service June 30, 1863, no further information, description on enlistment: 26 year old Merchant, born in East Tennessee, 5’ 9”, light complexion, gray eyes, dark hair

M258: Compiled Service Records of Confederate Soldiers Who Served in Organizations Raised Directly by the Confederate Government.

……

In the spring of 1864, Major General Samuel B. Maxey reorganized Confederate troops in the Indian Territory and created the First Indian Cavalry Brigade with Colonel Stand Watie as commander. The First Cherokee Regiment was part of this brigade. Colonel Robert C. Parks, commanding.

For additional information about the First Indian Cavalry Brigade, see the Encyclopedia of Oklahoma History and Culture article.
https://familysearch.org/learn/wiki/en/First_Cherokee_Regiment_(Oklahoma)_(Confederate)

……..

LIEUTENANT COLONELS. Thomas Fox Taylor, KIA 27 Jul 62, Bayou Menard (n. Fort Gibson); Robert Calvin Parks, replaced Taylor ( from Capt, 1st Co B) killed by a fellow officer in a personal difficulty at Fort Washita in April 1864; Clement Neeley Vann, succeeded Parks (prev pvt Co K, Drew’s Regt.?)
http://www.yvwiiusdinvnohii.net/history/CherConfed.htm

…..

See also: http://www.authentic-campaigner.com/forum/archive/index.php/t-7748.html