Taking a slightly different approach to that of AngryResident, the Clerk is your employee and it is up to you (ie. the council) to set the ground rules.
That said, I think it is worth remembering that a parish council is a public authority and - potentially - all emails are subject to the Freedom of Information regime.
On a practical level, my advice to my councillors (I've since retired) was "only commit to writing anything that you would be happy to see published on the front page of the local press". I realise that is a vast overstatement but it helped me to make a point and focus minds.