The council is governed by law regarding the advertising of vacancies. At the point of resignation, the local electoral officer at the Borough level should be informed and they will instruct the council to place an advertisement stating that electors have 14 days from the date of the notice to request a by-election. There have to be 10 electors writing to the election officer to trigger a by-election. If this is not achieved, then there is an obligation on the council to fill the position informally (a casual vacancy). There is no legislation as to how this should be achieved.
It certainly seems that your PC does not have a decent policy. In May 2019, the whole parish council lose their seats and they are all up for election. If you decide to stand then, and there are not enough candidates to call an election, you will automatically be elected. BTW, there is no requirement to co-opt in the 6 months leading up to the election in May.