Have you tried ordering his death certificate from the state archives? If the person supplying the information knew his parents' names, they should be on the certificate.
From census records (where he was listed as Dill F.), I'm assuming this is him:
Deal F. Pinell
Date of death 5/13/1958
Age 79 years
Place of death - East Baton Rouge Parish
Volume 7, Page 498 - 1958 Statewide Death Records
(I found it here:
http://tinyurl.com/LAdeaths)
It's also possible their names might be listed in his obituary. If you don't have a copy, you could try contacting the East Baton Rouge Public Library to see if it's possible to order an obituary for him. If you do that, I'd try the Bluebonnet Branch. Here's their web page:
http://www.ebrpl.com/LocationsandHours/bbr.html