I found your family in both the 1920 and 1930 censuses.
1930 Tarrant County, TX, Precinct 3, ED 114, Page 1 B.
W.C. McCaghren, head, 44, married, age 22 at first marriage, reads, writes, he and father b Alabama, mother b Georgia, Farmer
Lundy, wife, 37, married, age 15 at first marriage, reads, writes, b Texas, parents b Alabama
Roy, son, white, 20, attends school, reads, writes, b TX
Ana Mae, daughter, 18, attends school, b TX
Inez, daughter, 16, attends school, b TX
1920 Handley, Tarrant County, TX, ED 86, Page 13B.
Dwelling 269. owned farm (mortgaged)
Charles W. McCaghren, head, white, 37, married, reads, writes, he and parents b Alabama, Farmer
Lunda, wife, 27, married, reads, writes, b Texas, father b Alabama, mother b TX
Edna, daughter, white, 11, attends school, b TX
Roy, son, 10, attends school, b TX
Annie M, daughter, 9, attends school, b TX
Mildred, 6, b TX
Tom, brother, 25, single, he and parents b Alabama, laborer
And from the SSDI/Texas Death Index:
Tom MCCaghren, born 13 Nov 1894, d Jan 1987.Last Recsidence: Overton, Rusk County, TX.SSN 466-12-7477 (issued TX)
Mildred I. Starks, b 18 Jul 1913, d 23 May 1978 in Tarrant County.SSN 463-22-2282 (issued TX)
The death certificates for either William Charlie or Tom should give the names of their parents, to the best knowledge of the informants.Tom filled out his own Social security Application, and that should also list his parents.
Good luck with your search!