A parish Cllr is still a private individual and a private individual enjoys the privilege of freedom of speech.
If the comment posted is that of a private individual then it is a matter between the author and the subject which may also be subject to T&Cs of the social media platform.
If the comment opens with “I am a parish councillor” or “on behalf of the parish council” or if it unambiguously references PC business, then the author may have behaved in a way which could be of interest to a monitoring officer.
If the comment is on the PC Facebook page it could be more difficult, but not impossible, to argue that the comment is private rather than PC business, but you describe this instance as being “a community” Facebook page.
Cllr XYZ is bound by CoC and PC policies WHILST CONDUCTING PC BUSINESS but MR XYZ is just as entitled to express a personal opinion as the next man
You would need to provide more detail / context for the better standard of reply.