The only Donald Livermore birth from 1905 to 1925:

Surname   First name(s)   Mother   District  Vol  Page
Births Jun 1917   (>99%)
Livermore Donald Coleman Cardiff 11a  751

So is mother’s surname was Coleman.
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His parents’ marriage:

Surname   First name(s)   Spouse   District  Vol  Page
Marriages Jun 1914   (>99%)
Livermore William A J Coleman Cardiff  11a  962
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I can’t find his father as William A J in the 1911 census.

But there was a William A born in 1889 in Penarth, Glams which looks rather like him.  His father was Charles who was a cycle agent and born in Bristol, Glos in 1848.  Does not look a close relationship to Martha and Charles Livermore, millers of Cambs.
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The William A Livermores born from 1880 to 1900 were:

Surname   First name(s)      District   Vol  Page
Births Dec 1883   (>99%)
LIVERMORE William Arthur Wandsworth 1d 667
Births Jun 1888   (>99%)
LIVERMORE William Albert    Rochford 4a 414  <-- Essex
Births Jun 1889   (>99%)
Livermore William Alfred    Cardiff 11a  307

None of these are shown as William A in the 1891 census.
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Can’t find a death on the England & Wales indices for him.