Churchyards can create headaches for parish councils. Some have argued that it is unlawful for parish councils to pay for church graveyards at all, but most people think that they can.
The looming issue behind this question is that a churchyard can be declared closed, at which point the parish council becomes liable for its maintenance. It is possible to pass this responsibility off to the district council. But that isn't as easy a get-out as it sounds. The district council may not carry out the maintenance to the satisfaction of local people. Worse, the district council has a power to levy a supplementary charge on a parish that incurs exceptional costs, and so the parish might lose control but still finish up footing the bill.
If the churchyard is still open, then there will be revenue, and it would be reasonable to expect the church to help with the costs.
Most people regard a churchyard as a local asset, as an open space that is a valued part of the community.
In the end, it is a judgement on what is best for the parish, in the light of all the circumstances.