Have you ordered Emile and Anna's death certificates from the state archives? If the person who provided the information knew it, their parents' names would be listed.
Are either of these the correct people?
died 6/6/1929 in St. James Parish
(Order 1929 death record, Volume 26, Page 10636)
Mrs. Emile Theard
died 10/29/1935 in St. James Parish
(order 1935 death record, Volume 30, Page 12405)