When a councillor resigns, the parish clerk has to notify the monitoring officer of their resignation. A legal notice is then sent to the clerk which has to be published on the noticeboards, website etc. This notice give the 14 day opportunity for 10 electors to write in and call for an election. If requests are received after this time, they are not counted.
If the parish council 'decides' not to fill the vacancy, the monitoring office can insist that they do - there is an obligation on the parish council to keep the numbers up...