I don't know what source you used to look for SSDI information. Here is what I found:
A. Gilmore, b 22 Sep 1914[sic]; d Sep 1986; SS card obtained in LA; last residence Porter, Montgomery Co, TX. His SS number is also shown.
You can obtain a copy of the application he filled out to get his SS number. It should show the names of his parents, his residence at the time of his application etc.
Information on Death Certificates only as accurate as the knowledge of the informant.
There's a Gilmore family on the 1920 census Caldwell Parish, LA that has in their family a son named Wilburn, age 5, b LA. Can't read the mother's first name, and father is just initials, W.M. You probably need maiden name of mother anyway for DC. I'd go for the SS application. Alon Wilburn Gilmore should have known names of his own parents when he filled out that application.