Did not find his parents, only his wedding. There was a William T. Williams who served in the Union Military, out of KY in the 13 Regiment of the Kentucky Cavalry 1861-1865.
Kentucky, County Marriages, 1797-1954
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Citing this Record
"Kentucky, County Marriages, 1797-1954," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:V5ZC-GLG : accessed 14 January 2017), William Williams and Betsy Reed, 13 Oct 1805; citing Madison, Kentucky, United States, Madison County Courthouse, Richmond; FHL microfilm 183,305.