This is a tentative answer, but so far as I can see only in a situation where the parish council is in difficulties. For example, if too few members are left for the council to hold a quorate meeting, the District Council can step in. Or if the council fails to hold its first meeting after an election within the specified time, or if the council is in a procedural mess over electing new members.
My impression is that the District Council can only step in to get the parish council back into normal operation.
The exception to this is that if the parish decides not to have a council, then any responsibilities formerly held by the parish council are likely to devolve on to the district.
The District Council also has a Standards Committee that has a role in dealing with improper actions by councillors, but this does not involve the district taking over the parish council.