Section 84 of the Local Government Act states:
(1) A person elected to any office under this Act may at any time resign his office by written notice delivered—
(a) except in a case falling within paragraph (b), (c) or (d) below, to the proper officer of the council;
(b) in the case of a person elected to a corporate office in a London borough, to the proper officer of the borough;
(c) in the case of a parish or community councillor, to the chairman of the parish or community council;
(d) in the case of a chairman of a parish or community council or of a parish meeting, to the council or the meeting, as the case may be;
and his resignation shall take effect upon the receipt of the notice by the person or body to whom it is required to be delivered.
In other words, the chairman of a parish council may resign by written notice delivered to the council (not the clerk), and it takes effect when received by the council, presumably at a meeting.
The above does not apply to the vice chair, whose resignation takes effect when received by the clerk, creating a vacancy which must be advertised in the usual way to allow those on the electoral roll to call a public election to fill the vacancy if they so choose.