You didn't include any dates, or approximate dates, in your query.But, the name seems uncommon so I went looking.(^_^)
Combined info from TX Dth Idx 1964-1998 and SSDI:
Conley L. Necessary, b 5 Apr 1886; d 17 Feb 1974 Parker Co, TX; last residence Weatherford, Parker Co, TX; SS card issued in TX; single [can mean widowed, separated or divorced]
A copy of his death certificate should show the names of his parents.Or, a copy of his application for his Soc Sec card should show the names, plus his residence at time of application and other data.
Once you have the name of his father you can look at census records, beginning with the 1900.
I did not find Lucy, she may have died too early for these computerized indexes.
I did find them on the TX Birth Idx 1926-1949.They had twin daughters in 1931 in Tarrant Co, and 2 sons 1935 & 1941 in Bosque Co.Their names are spelled various ways, but I have seen this on the vital record indexes for TX.His middle name was apparently Lafayette.