That Eyre History site I linked to upthread is quite hard to navigate; it's difficult to get it to give you anything resembling a pedigree, but it doesn't seem to have a likely father for Edward. Still, it may be worth getting in touch with the site owner, because he's doing an Eyre one-name study and he's a GOONS member.
However, although I haven't explored it yet, this site looks as though it could be a real goldmine:
http://www.bradfieldarchives.co.uk/index.html
It doesn't seem to have parish registers online though, which is a shame.