Okay, looking at Lancashire BMD there are two possible birth regs for William's elder sister Martha in 1847 Colne sub-district, one with mother's maiden name Firth and one with mother's maiden name Whitham.
John Taylor married Jane Firth 30 Nov 1846 at St John the Baptist, Wakefield, Yorkshire. The image is on ancestry's West Yorkshire database:
http://interactive.ancestry.co.uk/22...nSearchResults
So it looks as though these are William's parents.