Finally spoke to the other twin over the phone and she gave me the name they believe is their father:
JOHN ATTEBERRY born 18 Sep 1902 possibly in Broken Arrow or Catoosa Oklahoma. When they were born he was age 40 and their mother was only 20. They have a census of 1930 which shows they had a sister 5 yrs older than them.
They also found the birth notice of their birth on July 14, 1945 mentioning their names as Eugenia and Yvonne Atteberry.
They think their mother was LILLIE BELLE SWADLEY born @ 1934 and that she died in 1974 in Colegate.