I think the short answer is a simple no.
My understanding is that CoC complaints must be passed onto your principle council, this is normally your borough council, and then dealt with initially by their monitoring officer. If you check your borough councils website you will hopefully find a policy about complaints against parish councillors.
Parish clerks or anyone involved in the parish council are not involved in investigating CoC complaints. The investigation of CoC complaints in undertaken by officers of your principle council.
If your clerk is actively investigating CoC complaints that have been made against a councillor I would suggest the council gets advice on your clerks conduct!