There isn't a legal obligation on parish council's to keep open footpaths, however:
The district or county (depending on your tiers) due to have a duty to ensure they are open, they can't turn a blind eye to one that is inaccessible / closed. (Though a lack of funds may delay their actions).
The parish council may have entered into an (in)formal agreement to survey their footpaths and ensure they are correctly recorded, our parish council was asked to help confirm the records held by the district and highlight and errors or anomalies on their maps.
A parish does have a legal power to signpost a footpath if the district or count doesn't.
However, the Parish has acted correctly to inform the district / county about a closed footpath (there is no duty to, just proper governance). The parish can request that it is formally closed, but they have no more statutory footing to request it than an individual to request it of the district/county.
The parish has not 'taken sides' in this dispute, they have acted reasonably.