From the Good Councillor's Guide
"Co-option: the council chooses someone to fill a vacancy if insufficient candidates are proposed for seats at an election. An ordinary election occurs every four years but there may be an election when a seat falls vacant at other times. In addition, if a vacancy occurs between elections (for example, by the resignation of a councillor), the council must generally find out if the electors want an election before they can co-opt. It is better for democracy if councillors are elected rather than relying on co-option, so they can be confident that the council is the community’s choice of representatives."
and The Essential Clerk Booklet
"In between elections councillors may leave for a variety of reasons. If a councillor decides to resign, they must write to the chairman while the chairman resigns by writing to the council. This creates a ‘casual vacancy’. You should contact the Returning Officer for guidance. First you follow proper procedures to find out if the electors want to offer a candidate for election (which may be contested or uncontested). If not, after the proper time has elapsed, the seat can be filled by co-option (provided that the four-year elections are more than six months away)."
It looks like the clerk at my parish council has not read either of these publications!