Is this them in 1891- moved to Sheffield?
It'll help to track him backwards.
Your best bet is to buy his marriage cert for £11 from the official GRO site and it'll tell you who his and Elizabeth's father's were.
England and Wales Census, 1891
Yorkshire, Yorkshire (West Riding)
Household Role Sex Age Birthplace
Edward Lingley Head Male 54 Lancashire, England
Elizabeth Lingley Wife Female 57 Cheshire, England
Albert Lingley Son Male 24 Lancaster, England
Arthur Lingley Son Male 20 England