I don't know if this helps, but there are other children born to this couple (with the spelling of the surname varying between WITE and WHITE).
This may give a clue to the names of Walter's parents:
MARY WHITE - Christening: 28 JUL 1776 Cholsey, Berkshire
JOHN WHITE - Christening: 14 MAR 1779 Cholsey, Berkshire
ELIZABETH WHITE - Christening: 07 JAN 1781 Cholsey, Berkshire
ANNE WITE - Christening: 02 FEB 1783 Cholsey, Berkshire
SARAH WHITE - Christening: 13 MAR 1785 Cholsey, Berkshire
CHARLOTTE WITE - Christening: 30 SEP 1787 Cholsey, Berkshire
THOMAS WHITE - Christening: 18 APR 1790 Cholsey, Berkshire
Kind regards
Philip Candy