It is not unusual for a clerk to provide that service for more than one council, as few clerkships are anywhere near a full time role. One of the many news items relating to the shortage of candidates in this May's elections included an interview with a vicar from Cumbria who mentioned in passing that he is clerk to eleven parish councils in his spare time! The contract of employment should set out what should happen in such a situation, assuming that your council was the first employer. There's usually a "must seek permission" or "must notify" clause, often with a statement that such a request would not be refused unreasonably.
Until recently, many clerks, especially in smaller parishes, used their personal e-mail address for council business. These days, with all parish councils required to use a website to make documents available to the public, there is an expectation that a dedicated e-mail account is used, not least for data protection reasons. This safeguard should, amongst other things, prevent members of the clerk's family from accessing confidential information and ensure that the e-mail records remain within the council when a new clerk is appointed.
It is unprofessional and unnecessary to use one council's e-mail account for the business of another council. If you have evidence that this is happening, an informal approach may be all that is required to rectify the situation. The chair or line manager of the clerk (individual or committee) should be responsible for determining whether to raise this formally in a meeting or informally in private.
In my opinion, the formal disciplinary procedure should be a last resort when all other avenues have been explored. Sometimes people are simply unaware that they have broken the rules. Your clerk might benefit from training on electronic communications, which your local association should be able to offer.