Question

can a member of the public start a vote of no confidence in a parish councillor

Answer 1

A member of the public cannot force an item on to the agenda, so a vote of this nature would require the cooperation of a councillor or the clerk.  The council could refuse to take the vote and agree a resolution to move to the next item of business.  A vote of no confidence has no binding consequences.

A member of the public can insist on a vote at a parish meeting, although again such a vote is not binding on the council.

Six members of the public can require a parish meeting to be held.  And then insist on a vote on any matter.

Opinions can be voiced, but there is no way to remove a councillor (unless they become disqualified) except at elections.  Having said that, substantial and continuing pressure from a sizeable number of the electors is likely to have an effect in the end.