I can't see why not. LGA 1972 section 214 first says that a parish council is a burial authority. Then it says that burial authorities may provide and maintain cemeteries.
You can find a detailed discussion of the terms churchyard, graveyard and cemetery at https://jakubmarian.com/difference-between-cemetery-and-graveyard-in-english/
. My reading of this is that there could be parts of the churchyard that are effectively a garden adjoining the church, but at least part of the churchyard is a graveyard and a graveyard is a kind of cemetery. Hence the parish council is, as a burial authority, empowered to maintain the portion of the churchyard that is a cemetery. It might be deemed hair splitting to separate out parts of the churchyard, and so better to simply maintain the whole area.
Some argue that only a minority are nowadays connected with the church, and it is unreasonable to put the burden on all the citizens of the parish. Against that, a churchyard is often a valued space within a community, and can be seen as a communal responsibility passed down from times gone by.
The provisions for closed churchyards are more to do with the facility for the church to hand over responsibility to the parish council. The parish council can, in turn, refuse to take this responsibility and pass it to the principal authority. However, there are drawbacks to this course. The principal authority may not maintain the churchyard to the satisfaction of local people, and it can also pass on the costs of maintenance (over which the parish would have no control) as a surcharge on the parish council. It is thus usually better for a parish council to shoulder the responsibility if it can.
There is an injustice here in the case of large churchyards in parishes with very small populations, but unfortunately I don't know of any solution to that.