This is awful and I'm really sorry you are going through it. The only thing I would ask you to keep in mind is that you don't have any concrete information other than other people's suspicions, so I think the only thing you can demand that your father speak to your mother about is just that, the fact that there is talk, that you have heard it, that you are upset. Even if there is a legitimate explanation (these may be rumors left over from inappropriate behavior that has stopped since your folks entered counseling) you should still insist that he talk to your mom about the fact that the rumors exist because otherwise you feel you are hiding things from her that she has a right to know.
If you are having trouble working up to it I might try something like "Dad, this is really awkward for me but everyone at your work seems to think you are obviously having an affair with OW because they say you (list of specific behaviors). Having this information puts me in an uncomfortable position because I don't want to interfere with you and Mom but I will not hide things from her either. Please talk to her about whatever is going on at work that is making people think you are not being faithful before someone makes a thoughtless comment and she gets hurt. I am not going to ask you what the truth is. I am, however, going to ask that, after you talk to Mom, you have her let me know. If she doesn't approach me before next Friday, when we have drinks, to tell me that you have talked to her about the situation, I will tell her what I have heard. Whether or not you speak to her before then is entirely your decision but the fact that I WILL talk to her about it if you do not, is mine, has already been made, please don't argue."
Then let them both know you only went to them because you felt uncomfortable with the information you had, that you don't want to be involved in how they are working this out, but that you won't take sides, which is why you needed the information you had out in the open. I would let DH and FIL know that you don't want to know anything else they hear, even though you really appreciate that they came to you with it this time, because it is too hard for you to have that sort of responsibility.
Wish I could come up with something better but I do agree you shouldn't try to keep this to yourself, for your own sake.