You didn't include any dates, or approximate dates, in your query, so I went looking at your message history to see if I could pick up anything more.I found enough that I think Henry and Mary probably lived long enough to generate death records.TX began keeping vital records in 1903 and you can obtain a copy of their death certificates, which should show the names of their respective set of parents.
I think you may have birth year wrong for for Robert William.Combined info from TX Dth Idx Online 1964-1998 and SSDI:
Robert W. Tune, b 8 Apr 1892; d 18 Aug 1974 Ellis Co, TX; last residence Midlothian, Ellis Co, TX; SS card issued in TX; single [can also mean widowed, separated or divorced.]
Your father is on the TX Dth Idx -- Robert L. Tune d 6 Nov 1973 Dallas Co, TX.He was too young to be receiving SS benefits, so he is not on the SSDI.
I'm thinking the Robert L. Tune on the SSDI might be the Robert Long Tune you mentioned as a cousin of your fathers.He was born about the same time period as your father:
Robert L. Tune, b 4 Dec 1920; d 19 Dec 2001; last residence Kerrville, Kerr Co, TX; SS card issued in TX. Happy Hunting.