What is the definition of exerting privilege when one is the third party in a divorce situation?

I am unaware of any 'third party in a divorce situation' privilege. I suspect it may only exist in Imagination land, if even there. Further, a third party in a divorce situation may be best served by exerting himself or herself as seldom as possible, at least for the time being. Trying not to assert oneself into the process is likewise a reasonable move.

If what you mean is that I am the other party to one of the party's infidelity, I am not sure that you have any privilege to exert. You are not the spouse.