There are connections between the 13th infantry regiment and Brown's 35th Cavalry regiment, that may have led to the confusion.
"Lone Star Regiments in Gray" says that 2 companies of Bates' regiment were transferred to Rountree's cavalry battalion in 1862, followed soon after by 2 more companies leaving under Lt Col Brown to form their own cavalry battalion.
Brown's 12th cavalry battalion and Rountree's battalion were later consolidated to form the 35th [Brown's]Texas Cavalry Regiment.