Since you know when and where Wm and Burton died, have you obtained a copy of their death certificates?They should show the names of their parents.
William Parrish, b 4 Jan 1902; d Nov 1962; SS card issued in TX.
Combined info from TX Dth Idx 1964-1998 and SSDI:
Icy Dora Parrish, b 6 Jan 1901; d 20 Jul 1998 Kaufman Co, TX; last residence Kaufman, Kaufman Co, TX; SS card issued in TX.[listed as Dora Parrish on SSDI]
Burton R. Parrish, b 14 oct 1915; d 27 Jan 1977 Val Verde Co, TX; last residence Meadow, Terry Co, TX; SS card issued in CA.[SSDI shows birth year as 1914]
Burton Raymond Parrish and Arvanell/Auvanell Caswell had a dau born 1944 Lynn Co, TX, and a son born 1948 Terry Co, TX.
Accuracy of death certificates only as good as the knowledge of the informant, but all these people filled out their own application for a Soc Sec card, and they should show the names of their respective set of parents, residence at time of application, and other data.
Using online directory, there is a Richard Parrish listed in Meadow, TX.Probably that son of Burton's.