Councillors can write letters or emails that show their role. Personally, I use a different email address for such matters, and include a standard signature block with contact information and the title "XXX Town Councillor". (My council chooses to be called a Town rather than Parish, although there's no legal difference).
The area where this gets tricky is writing to the press or any similar public statement. In that case, if someone identifies themselves as a councillor, they must make it clear that they are expressing a personal opinion and not speaking for the council (unless they have been authorised to do so by a decision of the council).