Ugh. This one is a messy issue. Ideally, talking with the other girl's parent about it first would have been the best. The kid got that revenge strategy from somewhere sourced to an adult. They don't make up child abuse on their own. However, parents tend to be extremely hostile when told their kid is being bad, almost like an automatic defense mechanism.

The problem here is that the father could be incorrectly accused of child abuse or worse and investigated and life turned upside down because a kid contrived a fake story. As the mother, you're probably in the strongest position to force the truth of the matter because no one will believe the father once accused or until proven innocent by a lengthy investigation and damage.

On this one, the best course right now is to cut off communication with the other girl and her parents entirely. Its better for things to be awkward than to spin out of control with further involvement. The school psychologist is a mandated reporter, regardless of being a counselor, so there might be fallout there still down the road. Investigators are required to follow leads, irrespective of whether parents think the whole story is made up. So be prepared for that.

I saw this same scenario play out with parents in my kids' school. It didn't get to an investigation, but the issues were similar in terms of events. They arrived at the same conclusion of a complete cutoff, even if the kids didn't understand why at the time.