CFI pilot reported a NMAC while departing an airport using parallel runways. The intruding aircraft drifted off course and towards the instructor. The instructor does not think ATC understood the events that were taking place because when questioned; the controller said all aircraft were in their correct flight path.

Narrative

I had just done a touch and go on a flight with another CFI and we were climbing out of Runway XXR at ZZZ. As is common; there was traffic taking off at the same time from the parallel runway; XXL. We had them in sight and their climb performance was a little better than ours; so they were consistently about 100-200 ft. above us. We were told to extend upwind by Tower; so we were simultaneously watching traffic and extending upwind. We were partially overtaking the traffic; even though they were above us; but all of a sudden we lost sight with them and realized it was because they had passed directly over top of us. We double-checked and were still on the extended centerline; but they had drifted to the right. I was sitting right seat and looking out my window and could see them right behind us. At that point; Tower called our crosswind and I told them that our parallel traffic had crossed over top of us and was interfering with that. Tower said that according to the radar they were below us and were no factor. I stated that I saw them out to our right behind and above us and Tower claimed that looking out the window; they were to our left and below us. I believe that Tower actually mixed up the two planes because of how far that plane had drifted to the right. Eventually we kind of held level and they continued climbing since they were departing so I was able to make my normal crosswind and we came around and had a normal landing. It was concerning one; that the traffic was able to drift so far to the right without making any corrections; but two; that Tower told us everything was fine. If we hadn't had the situational awareness that we did and hadn't been watching that traffic; we could have easily turned into their path.