There is a wonderful article about finding records in Louisiana at http://freepages.rootsweb.com/~cmbs/genealogy/larecords.html
I don't know if Virginia was recording births in 1804 or not. You can look that up
I found their marriage license but no parents names were on it. I'd look for the marriage announcement in the local newspaper.
Louisiana wasn't officially recording deaths in 1854 except for probate records.
I'd contact the following and ask if they have a local death index, indexed newspaper death notices and cemetery inventories.
DeSoto Historical Museum, http://desotohistoricalsociety.com/Museum.htm
DeSoto Parish Library, www.desotoparishlibrary.org/services-and-programs/louisiana
If they were Catholic you might find their parents in Catholic church records. There is a good article about how to find these records at https://freepages.rootsweb.com/~cmbs/genealogy/larecords.htm...
Another option is to look at a family tree and find proof that the information there is correct.
John Hiram Williams father was John Williams, b. 1785, VA and died 1860 Pope Co. Illinois. His mother was Mary Coleman, b. 1789 VA and died 1848 in Pope Co. IL
Source: Fitz/Merritt Family Tree
I'd look for John (John's father) probate records and John Hiram's name in those records. Successful Searching!