I would suggest that the term 'Local Authority' in the sense of the Anti-Social Behaviour, Crime and Policing Act 2014 refers not to Parish and Town Council but Borough and District level councils.
Correct me if I am wrong but our Borough Council has the powers of enforcement, not parish councils. We can create a byelaw for land that we control but it is a civil matter, not a criminal one.
Edit - I believe I am right in my assumption as the link provided by Samuel states "This covers district councils, London Boroughs, county councils in an area where there is no district council in England (along with City of London and the Council of the Isles of Scilly) and county councils or a county borough councils in Wales"