That is an interesting question. I have not come up against that issue, so I am not entirely sure whether the investigation would show up on background checks or not. Let me clarify that I am only referring to specific checks for CPS history here. I am sure there are certain types of background checks in the real world that have nothing to do with CPS and do not look for that information.
My initial thinking would be to direct you to my May 25 post regarding removal of role information. Even if the parents were not designated as perpetrators, I do not think it could hurt to send in an immediate request to remove all role information where the parents are concerned. I have never heard of a case where there was not an alleged perpetrator, so the parents need to be very careful that they were not given that label.
As far as the child is concerned, I do not think that the fact that the child was an alleged victim in a CPS case can or will be removed from CPS records. Many times I have seen CPS run a parent though their system and it is revealed that the parent was an alleged or confirmed victim as a child.
The best answer is that I am not sure, however the above might point you in the right direction. Take a look at that old post, too.
Christina M. Davis