At the time that of John H. Parker's death the 24th Arkansas Inf. was being organized from the "extra" soldier of the 19th Arkansas and 33rd Arkansas at Camp White Sulphur Spring, near Pine Bluff. In August 1862 the camps at White Sulphur Springs were being sweep by Measles, Pneumonia, and fevers, mainly brought on by exposure and poor living conditions. Many of the men who died there as a results were never "Officially" enlisted in the 24th and had been dropped from the rolls of the 19th as they were never really an "official" part of the 19th regiment either.
It appears that Levi was a part of the 24th Arkansas that was stationed at St Charles, Ark that did not make the march to Arkansas Post just in time to be captured when that post surrendered January 11th, 1863. As a consequence he was reorganized into Hardy's Regiment.
As I said before in May of 1863, because of the fact that Hardy's Regiment for a time, during the Battle of Little Rock, it was assigned to Gen Daniel Frost Missouri Brigade some of what little records of the deaths from the 19th and 24th Arknasas exist (mostly just names) appeared in Missouri archives records as deaths of Missouri soldier. You might be able to check the Missouri Adjuntant General report of names of Missouri soldiers that dies during the War to see if John H. appears in that list. Because most of those names do not have regiments listed with their names they do not appear in other lists of muster rolls. I would check my copy but it is loaned out and has not returned in several years.