Look at the Social Security Death Index at ancestry.com
I found 8 individuals named Joseph Fuselier.
I identified the parents of Joseph Fuselier born 6 Nov 1907. He was bt. Joseph Isom Fuselier in Sacred Heart Catholic Church, Ville Platte, LA. His parents were J. B. Avie and Temille Fontenot.
I identified the parents of Joseph Fuselier born 24 Dec. 1897. He was bt. Joseph Fuselier in St. Martin of Tours Catholic Church, St. Martinville, LA. His parents were Rodolphe & Adrienne Duchamp.
I identified the parents of Joseph Fuselier born 1 Jan. 1885. He was bt. Joseph Dorano Fuselier in Our Lady of Mt. Carmel Catholic Church, Chataignier, LA. His parents were Damon and Delphine LaFleur.
I could not identify the parents of Joseph Fuselier born 24 Dec. 1907.
I could not identify the parents of Joseph Fuselier born 7 Mar. 1905.
The other Josephs were born in 1925, 1927, and 1932. My sources go only to the year 1909 for Southwest Louisiana.
As a rule, I do not find middle names listed on the SSDI. Sometimes the individuals middle is the one he used when he applied for his SS No. I also checked for Elba Fuselier and for his wife Rita Fuselier. Found neither. SSDI uses the wife's married surname.
Even if your ancestors did not pay into Social Security, if they were still alive when Medicare was implemented in the early 1960s during the Johnson administration, they were assigned a SS No. so they could receive Medicare benefits.
When my 91-year-old-mother died this year, SS did not pay the $255 death benefit because she never paid into SS. She collected benefits on my father's earnings. However, her name is listed on the SSDI. This is good to know.
If you would like to know the parents' names for the Joseph Fuselier individuals I could not identify, you can click Request Information beside their names. A form letter appears that you can mail to the SS Adm. and SS will send you a copy of their original application that contains the names of their parents and other valuable genealogical information.
You say your ancestor lived in Acadia, LA. Is that Acadia Parish? In Fr. Hebert's "A Guide to Church Records in Louisiana" there are many churches listed in Acadia Parish. Perhaps your ancestor was baptized at one of these churches.
Church Point Our Lady of the Sacred Heart (est. 1883) 114 N. Bienvenu St., Church Point, LA 70525
Crowley St. Michel the Archangel (est. 1895) P. O. Box 406, Crowley, LA 70526
Iota St. Joseph Church (est. 1883) P. O. Box 8, Iota, LA 70543
Mermentau St. John the Evangelist (est. 1910 P. O. Box 176, Mermentau, LA 70556
Mowata St. Lawrence Church (est 1905) Rt. 2, Box 33, Eunice, LA 70535
Rayne St. Joseph Church (est. 1972) P. O. Box Rayne, LA 70578
Roberts Cove St. Leo Church (est 1883) Rt. 1, Box 23, Rayne, LA 70578