Jane Bourne's birth certificate gives her parents as John and Mary Bourne, formerly Vaughan. Jane was with her mother in the 1851 Census and with both her parents in the 1841 Census.
In fact there are 10 children listed in the 1841 Census.

The fact that Jane's parents were John Bourne and Mary Vaughan seems fairly certain.

John was apparently 18 years older than Mary. A possible marriage has been found.
Both were listed as widowed.
  

Based on John's age at the 1841 Census he was born about 1783. The age is normally rounded but he is listed as 58.
However, the 1841 Census seems to say that John was not born in Warwickshire.

If John had been born in Warwickshire there are three possible baptisms, the first of which the most likely.
20 Sep 1785 at St Martin Birmingham to Richard or Robert Bourne and Jane (transcribers differ).
24 Feb 1789 at St Martin Birmingham to Penson Bourne and Elizabeth
10 Apr 1780 at St Philip Birmingham to William Bourne and Catharine
  

The 1851 Census shows that Jane was widowed. Therefore John died between 1841 and 1851.

A search of ancestry revealed three possible entries in the BMD that could fit.
1846 Q4	Birmingham	v16	p204
1846 Q4	Birmingham	v16	p238
1848 Q4	Birmingham	v16	p220

As indicated there are two separate entries in 1856-Q4.
Neither Ancestry nor Freereg showed any burials matching these three deaths.