Hopefully a rootschatter will be able to search the Bromeswell parish records for you - alas, not on the IGI.
Looking at this family on the internet and NBI think that Elizabeth is probably the daughter of George King who married Sarah Green 26/2/1794 Marlesford. They had a dau. Elizabeth bp.10/3/1794, bur.30/3/1794 at Marlesford but no other children there. Think they then went to Bromeswell.
From submitted trees looks like:
John King bc.1736-40 bur.13/5/1790 Marlesford married Elizabeth Markin (bc.1736, bur.10/1/1817 Marlesford) married 14/11/1757 Marlesford. Children baptised Marlesford were:
Catherine bp.23/4/1758
Martha bp.28/10/1759
Elizabeth bp.18/1/1761
Susanna bp.19/1/1763, bur.30/1/1763
Mary bp.5/1/1764, bur.29/1/1764
John bp.12/11/1765 bur.8/7/1828 Bromeswell
Charles bp.30/5/1767 bur.15/4/1770
Susanna bp.5/1/1769
George bp.22/3/1771 bur.27/8/1848 Bromeswell
Henry bp.30/7/1772 bur.23/8/1772
Charlotte bp.29/6/1774 bur.7/8/1774
Joseph bp.9/4/1778 bur.15/2/1852 Bromeswell
William bp.9/4/1778 bur.6/1/1822 Marlesford
Three of their sons went to Bromeswell - already mentioned George who married Sarah Green, Joseph who married Elizabeth Burroughs/Burrows 12/10/1803 Woodbridge and John who married Deborah Pannifer 24/11/1807 Bromeswell.
I note that Elizabeth Lovett (nee King) was buried Bromeswell 11/8/1833 aged 32 - also a George King Lovett, infant, on 17/10/1832 and a Sarah Lovett, 1+, 9/3/1827.
From the 3 sons who went to Bromeswell 2 didn't marry until after 1801 (and Joseph actually had a dau. Elizabeth later) so George King and Sarah Green look the likely parents. Hopefully someone will be able to confirm this after a check of the parish registers.
Good Luck.
Annette