Hi Nicola
Welcome to Rootschat. The first thing you need to do is change the @ in your e-mail address to [at] or you will get loads of spam.
There is a marriage (on Familysearch.org) of a James Pemberton to an Elizabeth Dickenson on 08 November 1801 at Manchester Cathedral. It is an extracted record, which means it has been extracted from the Parish records, and is probably correct. Unfortunately, marrying at Manchester Cathedral doesn't prove that either James or Elizabeth were born in Manchester. People came from all over the area to get married there. I have ancestors, who lived in Cheshire who were married there, as well as a couple from Liverpool.
There is a baptism of a Francis Pemberton, at Manchester Cathedral with parents James Pemberton and Elisabeth on 8 May 1808 not 1809 as you said. Again it is an extracted record, so probably accurate, although I suppose the year could be wrong if someone couldn't read the parish records properly.
Unfortunately, there are too many options of births of James Pemberton and Elizabeth Dickenson to decide which, if any, are the correct ones.
Lizzie