Was Sarah a spinster or widow when she married R Shackleton? Could it be that she was married to a Greenwood before Shackleton and that she was actually born as Whittaker? (Making son a legitimate child of Greenwood but that he took on his stepfathers name of Shackleton in later years?)
It's worth having a look at
http://www.lancashire.gov.uk/education/ ... alogue.aspJust type in the surname of mother or child or suspected father or try them all! I don't think all of the records have yet been catalogues but if there is an entry in the catalogue, the Lancashire Record Office should have it and you may be able to request a copy for a small fee. If you request a copy, be sure to quote any reference numbers you find as well as their description.