Section 4(1)(ii)(c) of the Parochial Church Councils (Powers) Measure 1956 provides that a Parochial Church Council is responsible for the maintenance of a churchyard.
The Diocesan website has a page of documents about Church Buildings and Graveyards, which includes guidance on the testing of unsafe memorials: ‘Churchyards and Memorials‘. The Government has also issued guidance on testing for safety of memorials, Burial grounds: guidance on managing unstable gravestones.
There is a procedure for transferring to a Parish Council or District Council the responsibility for maintenance of a churchyard which has been closed for further burials by Order in Council (see below).