Elizabeth HARRISONS baptism on 15 Aug 1802 shows her mother as ELIZABETH PLACE. Her father Thomas HARRISON with the occupation mariner.
There is a note transcribed as 3rd daughter" " N. Of Ryton
Thomas HARRISON and Elizabeth PLACE were married 7 June 1787 at St John the Baptist, Newcastle upon Tyne.
From a quick search (not extensive), I can find the following children for Elisabeth and Thomas HARRISON (a Mariner):
Margaret - bapt 23 Sep 1787 Tynemouth
Martha Born 28 June 1791, but bapt 8 Dec 1793 Newcastle upon Tyne (baptised same day as brother John)
John born 15 Sept, bapt 8 Dec 1793 Newcastle upon Tyne
Thomas born 5 April, bapt 10 July 1796 Tynemouth
Elizabeth born 2 March, bapt 15 Aug 1802 Tynemouth