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I assume that you are referring to a police detective (and not a private detective). I would simply suggest that you and your wife ask him to not call you further. If he has credible evidence of a crime, then he must of course investigate that. (In addition he may have a duty to notify DYFS. If he has not so notified them, then that may suggest that he does not find these statements credible.) Even if your son had a knife, that does not seem highly relevant. You could ask to listen to the tape recording, but it is not clear what that would resolve in any event.
If the detective continues to harass you, then there are administrative procedures you can use to report him to his supervisors. The issue will then be whether he is legitimately investigating a crime. At some point, without any new substantial evidence, the detective should either commence a formal legal proceeding, or back off. You may also theoretically have claims against the boy's father (e.g. defamation, abuse of process), but it may not be worth pursuing those.
Also, your decision to not speak to the detective may be a good one. Many times a person makes a statement which seems innocent, but which the police can later use against the person.
I would hope that the police would not be focused on such an old criminal offense (30 years ago). But you make a good observation about the reference to the knife; it is conceivable that they are thinking about this. But if your record has been clean for 30 years, and there is no real evidence of neglect or abuse (as determined by DFYS), then they should just leave you alone. Regarding the Miranda issue, perhaps the detective was just being very cautious. (I am not even sure that you were in "custody," such that he might not have been required to "Mirandize" you anyway.) I would suggest that you play it by ear at this point. Hopefully too they may realize that the father is unstable, especially if he is not taking his psychiatric medications. It is always difficult to prove your innocence in this sort of situation; i.e. proving a negagive; but you also of course want to keep thinking of anything that would exculpate you if they were to come back with some charges.