A parish councillor cannot, unless authorised, by a vote of the council, claim to represent the Parish Council on an issue.
However, they are parish councillors and can title themselves as such.
For example, I assist parents with school applications appeals and use my title, but my letters always have a disclaimer that the letter is personal and not the opinion of the Parish Council.
The council may adopt a policy to restrict the use of title if they wish to do so.