There is no guaranteed way to do this. It is possible for parishioners to insist on a parish meeting, and that meeting can take votes and could pass a vote of no confidence in the parish council and ask that it be dissolved. However, this request has to be put to the district council, which is likely to be reluctant to act while there is a council in place.
There have been instances of a parish council (such as Radnage) being dissolved as a result of pressure from a parish meeting, but the actual mechanism for this was that the parish councillors were eventually persuaded to resign en masse. When there are no parish councillors left, cooption becomes impossible (as does anything else) and the district council is obliged to step in and arrange an election.
Sometimes public pressure (perhaps including a parish meeting) will result in sufficient resignations to change the character of the council. Whenever there is a vacancy, an option exists for ten electors to demand a poll, rather than cooption. The cost of the poll will fall on the parish, as it does if the councillors all resign.
The simplest approach, especially with the next election being only six months away, is to ensure that new people stand for election as councillors in the elections that routinely occur every four years. The parish does not have to pay for the four yearly election. EDIT: I was mistaken on this last point. Principal authorities can charge back election expenses to local councils. Many local councils have their elections at the same time as the district council elections, and in this case, the most that can be charged is a pro rata portion of the total election cost. The administrative cost of charge backs may not be worthwhile. In the case of the council on which I sit and many others, the district council has always waived the charge.
It is possible for a parish council to be permanently dissolved, leaving all local matters to the district council. Again, this is usually instigated by the sitting councillors. It is usually better for a parish to try to find councillors who can work in relative harmony with local people rather than abandoning parish initiatives entirely.