From censuses it is apparent that Sarah Oliver was born in Hoo, Kent.
This could refer to St Mary Hoo or Hoo St Werburgh which are a few miles apart.

There is a baptism for Sarah Oliver at Saint Werburgh Hoo in 1787 which gives her parents as Thomas and Mary Oliver.

The identity of Thomas Oliver as the father of Sarah therefore seems fairly certain.
  

The following are baptisms of children to Thomas and Mary Oliver in Kent
Sarah was baptised in 1788
Thomas - 15 Aug 1780 Snodland, about 9 miles south of Hoo.
William - 28 Aug 1786 Maidstone, about 12 miles south of Hoo
Elizabeth - 28 Aug 1789 Maidstone
James - 21 Mar 1790 Chatham, about 5 miles south of Hoo
    

        When I (David Phillips) was younger we had a Dictionary inscribed
        Mary Oliver  MDCCLVIII
        The year is therefore 1758. This is very unlikely to be Mary Oliver the mother of Sarah because of the age (Sarah was born 30 years later). But it could be the mother of Thomas.