Since you know when and where William died, have you obtained a copy of his death certificiate?It should show the names of his parents.
You said you found William on the 1920 Ringgold Co, IA, on the same page is:
Burch, Chester, h, 57, md, b IL, KY, IN " Elenor, w, 49, md, b IL, Maine, IN " Howard, s, 27, b IL
1880 census Benton, Benton Co, IA:
Category: Name;Relationship to head of household; Marital Status; Gender; Race; Age; Birthplace; Occupation; Father's Birthplace; Mother's Birthplace.
Herietta BIRCHSelfWFemaleW41INKeeps HouseNYNY Chester BIRCHSonSMaleW17ILLaborerINIL Siam BIRCHSonSMaleW15ILLaborerINIL Charles BIRCHSonSMaleW12IAAt HomeINIL George BIRCHSonSMaleW10IAINIL Henry BIRCHSonSMaleW6ILINIL