Combined info from TX Dth Idx Online 1964-1998 and SSDI:
Alvin Rudolph Biggs, b 28 May 1907; d 17 Aug 1968 Tom Green Co, TX; SS card issued in TX.
Thelma Louise Biggs, b 21 May 1911; d 13 Mar 1996 Tom Green Co, TX; SS card issued in TX.
There are 2 Clarence Biggs on SSDI that died in 1965.Was that Las Vegas, NV; or Las Vegas NM?
Clarence Biggs, b 21 Oct 1890; d Aug 1965; SS card issued in NM.
Clarnece Biggs, b 3 Jul 1905; d Jun 1965; SS card issued in OK.
Death certificate for Alvin should show the names of his parents.Or, a copy of his application for a Soc Sec card, which he filled out himself, also should have the names of his parents.Once you have his father's name, check out census records.The 1910 census when Alvin would be about 3 yrs old.If there are older children in the household, check out the 1900, it asks some good questions.Then follow up on the 1920, too.
Using an online directory http://www.four11.comhttp://www.four11.com there are 10 Brent Biggs. You might try dropping a line to each of them.
If the wife of your Clarence was named Grace, they also had a son named Sydney.There are 2 from the online directory named Sydney Biggs.