Other than locating his WWI draft registration, I was not able to dig up anything definite. However, his name suggests he is descended from William E. Gilliam b. 1826 in Scott Co., VA. William E. had 5 sons: John B., Stephen, Ira, William Patton, and Martin L.
William's draft registration shows that he was supporting his mother, so this suggests that his father either left or died before 1917-1918.
Of the 5 sons of William E. only William Patton fits this pattern. John B. d. in 1942, Stephen in 1856, Ira in 1937, and Martin in 1925.
There is a W. P. Gilliam listed in Wise Co., VA in the 1900 census. The household shows:
W.P. May 1869 40 Married 21 years VA VA VA
Anne E. 1862 38 gave birth to 9 children and 9 still living VA VA VA
F.M. son Nov 1880 19
Dora dau 1882 17
W.C. son Nov 1887 12
G.F. son Nov 1887 12
W. A. son 1891 9
Ida W. dau 1893 7
Cordela dau April 1897 3
Clara dau April 1897 3
Flora E. dau Feb 1899 1
Of the sons listed above, W. A. is the closest in age. Census data is not always accurate so this could be William E. (My great-grandmother is twice listed as Augustus A., male.)
1910 Wise Co.
Annie Gilliam 49 Widowed had 9 children, 8 still living
Ida M 17
1920 Wise Co.
Anna E. Gilliam 52 widowed
One way to check on William's parents would be to order the death certificate for him. Since he died in 1956 it should be available. As long as the person who provided the information on the dc knew the names of the parents, it should give you the names.