I searched for Walter and Sarah in Milam and Limestone Counties, Texas but didn't find them.
Born: 1841, Ohio
Sarah Jane Markham Johnson
Born: 1845, Mississippi
Source: Markham-Barber Family Tree, ancestry.com
Most larger public libraries provide free access to Ancestry if you have a local library card.
This tree said they lived in Limestone Co from 1860-1880 but I couldn't find them or their son Lee.
1900 Houston, Harris, TX
William G. C. Johnson, b. OH, Oct 1840, bricklayer
Sarah J., wife, b. MS April 1844, had given birth to one child who was alive in 1900
Antonette Z., dau, b. TX Feb 1877
Then in 1910 in W G C and Sarah were still living in Houston. The family tree above shows they were living in Houston then but I don't think W G C is your Walter.
I saw your family tree and you have that Walter and Sarah were living in Houston in 1910. You might want to look at the William G C and Sarah J living there in 1900.
Does Lee or H. H.'s obituaries mention their parents? Are they included on their death certificates? They might not be listed because that info is provided by the Informant at the time of death and sometimes they don't know the parents' names. Successful Searching!