In some respects, the answer is in the question; when someone writes to "the parish council" . A parish council is a separate legal entity in its own right which is separate from individual councillors. This is very much like a business which is separate from its owners or directors, albeit that it is the owners/directors/staff who will deal with issues raised by any correspondence. Putting this into a town and parish council context, correspondence is therefore technically addressed to the proper officer who is the legal entity responsible for receiving it on behalf of the parish council and your internal systems will dictate how that then is dealt with.
If parishioners wish to write or contact parish councillors as individuals rather than the council as a whole, then clearly they can do so and many councils publish their councillor contact details separately from the "corporate" address to facilitate this but everyone needs to be clear that the official lines of communication to "the parish council" is via the Clerk who will then act in accordance with the internal systems and controls that the council has agreed and which are consistent with all sorts of legislation which of course would include data protection regulations.