His WWI Draft Registration Card filed in Cleburne County has Walter Stith Simmons b. Feb 28, 1891 in Cleburne County AR, farming, supports mother, wife and child, age 25 and living in Pearson, AR.
The family is in Cleburne County censuses from 1900 to 1930.
A Walter S. is in widowed mother's household Lavisa E Simmons, in 1900 AR Cleburne CO Rimy Twsp. He has older brothers born before 1880, Earnest R and George L.
So look at 1880 census for Gray Twsp, White County AR Simmons family. The 2 brothers of 1900 are in the household of Dekalb (or De kalb) and Levicy Simmons.
Arkansas Death Index has Lavicy E Simmons died Cleburne Co. March 23, 1928. I do not find Walter and death 1935, but I do fins a W.S. Simmons died Poinsett Co. Jan 28, 1939.
Order death certificates for confirmation.
There is a family tree on Ancestry using the name Walter Stith Simmons and has birth date same as WWI card and death date and place same as Death Index and also has a picture of him as a adult. Married Nancy Coe and there is a picture of her later in life "Grandma Simmons" written on bottom.
Looks like James Dekalb Simmons and Livinia/Livicy Elizabeth Redus as parents.