I found this information to back up what you have already:
Source:Louisiana Statewide Death Indices 1945-1949
David Hugh Allen, died 19 Dec 1947 in Vernon Parish, LA, age 69
Race:White or Mexican
You should order David's death certificate from the state.It will likely list the names of his parents, to the best knowledge of the informant.
Is it possible that John Arthur and Daisy married before 1936?I found this entry in the 1930 census which sounds a little like them:
1930 LA, Vernon Parish, Police Jury Ward 2, ED 8, Page 10B.
Dwelling 199/Family 199.
Arthur Allen, head, rents home, owns a radio, white, 22, married, age 21 at first marriage, reads, writes, he and parents b LA, farmer
Daisy, wife, white, 16, married, reads, writes, she and parents b LA
Here is your family on the 1920 census:
1920 LA, Vernon Parish, Police Jury Ward 2, ED 94, Page 1B.
Dwelling 21/Family 21.Lives on farm
David Allen, head, rents, white, 42, married, he and parents b LA
Ola, wife, white, 32, married, she and parents b LA
Eddie, son, 16, b LA
Liddie, daughter, 14?, b LA
Arthur, son, 13, b LA
Lizzie, daughter, white, age unreadable, b LA
Setase?, son, white, 6?, b LA
Unice, daughter, white, 2, b LA
Lureye, daughter, white, 2, b LA
And this is a very BIG guess for David's parents.I searched the 1880 census everyname index (available free at www.familysearch.org) and found only 1 David Allen living in Louisiana who was under the age of 5.Your David's death certificate could confirm this:
1880 LA, St. Martin Parish, ED 34, Page 2B
J.W. Allen, white, male, 30, head, divorced, swamper, b LA
Louise, white, 17, sister, keeping house, b LA
Mercedes, white, 10, sister, at school, b LA
Oscar Allen, white, 28, head, swamper, b LA
Mary, white, 26, wife, keeping house, b LA
David, whie, 3, son, b LA
Helene, white, 1, daughter, b LA