An instructor pilot reported the Tower Controller's instructions to their aircraft and an aircraft departing a parallel runway caused an NMAC.

Narrative

My Student and I were flying the pattern on Runway XXL at ZZZ. We had already done a number of takeoffs and landings (some full stop; some touch-and-gos). We had just taken off from XXL and were climbing when tower asked us if we had the aircraft just to our right in sight. We answered in the affirmative; that we had the traffic insight. A note about flying at ZZZ: Runways XXL and XXR are very close together; so it was not particularly unusual that there was an aircraft just off to our right; at our same altitude. But it is also very important that each aircraft stay 'on their side' and out of the others' pattern space.The Tower then responded by telling us to the best of my recollection 'That aircraft will be making a left crosswind turn in front of you; follow that aircraft for left traffic; RXXL.' The problem was that there was no room for them to safely make such a turn. But only a moment later; seconds; if that the other aircraft in front of us proceeded to turn directly into us. I grabbed the throttle; reduced power; and stopped our climb immediately and even descended a small amount (we were too close to the ground to safely descend any appreciable amount). The other aircraft was now directly above us. It then changed course again and assumed a position directly above us and now on the same upwind course; climbing as if they had departed RXXL. We could see the aircraft right there in front of us; seemingly only a small number of ft. above. All of this started while we were still very low; I'd estimate around 300 feet AGL. Normally I would have descended in such a case; but given our low altitude that wasn't an option. We also couldn't turn right due to that being the climb path for other aircraft on RXXR; and we couldn't turn left since the indication was that the other aircraft was supposed to be turning in that direction.The Pilot of the other aircraft appeared to be blissfully unaware of our presence and appeared to make no effort to avoid or acknowledge us in any way. This entire event was very unsafe; putting both aircraft very close together for an extended period of time. From my perspective; this entire event was precipitated by the tower's unsafe actions: (they = Tower) They turned the other aircraft directly in front of us; without adequate separation; and for no good reason. They should have turned us first; if anything. Although they asked us if we had the other aircraft in sight; they did not ask if there was room for them to safely turn in front of us. I also don't remember them asking the other aircraft about us; or telling them to maintain separation but it's possible that I didn't hear it since it was not my call sign. They started this whole event while we were still very low and slow; and with few if any good options. Even after the event started; they failed to issue any further alerts or instructions to either aircraft to resolve the hazardous situation that they put us into. We had to have been showing as (nearly) one aircraft on their display (or out the window) since we were so close together.I've asked myself 'what could I have done differently?' But I am at a loss to change anything that I did. I kept us low enough not to be hit by the aircraft above; yet high enough not to hit anything below. I stayed 'in our lane' to avoid interfering with other traffic on XXR. And I avoided the very real possibility of the other aircraft turning left into us. Again; I am not sure what I could have done differently that would have been any better than what I did. And we avoided a collision; so what I did worked out OK. Could I have said anything on the radio that would have helped? Again; I am not sure. The tower was talking to a number of other aircraft at the time; so I would have been fighting with them for air time; which is never a great choice. And it is not clear what I should have said; if anything. Again; the Tower put us in a very unsafe situation with very few if any options. If I could havemaneuvered the aircraft in some way to improve the separation I would have done that and then informed the tower of what we were doing and why. But with no options to move I didn't really see what I could have said that would have helped to resolve the problem in the time frame necessary.