C172 Flight Instructor with student reported a NMAC when another C172 departing a parallel runway turned into their flight path. Flight Instructor took evasive action to avoid a collision.

Narrative

My student and I were taking off after doing a touch and go on Runway XXL at ZZZ in Aircraft X. We were in the upwind and climbing up to pattern altitude for left closed traffic for the same runway. Before the departure end; there was another C172 abeam and above us; Aircraft Y; who checked in with the touch and go runway Tower on XXX.XX that they were runway heading off of XXR. Tower advised Aircraft Y that 'traffic [is] ahead and low to your left; there Cessna; almost abeam ya. That's the traffic that I want you to follow for left closed traffic XXL.' Aircraft Y responded 'XXL cleared; uh; we have traffic in sight Aircraft Y.' Tower replied 'Aircraft Y; follow that traffic and left closed traffic.' Aircraft Y read back to Tower's response with 'left closed traffic; Aircraft Y.' During this time; I was telling my student that Aircraft Y was talking on the frequency and how we don't have them in sight due to the right wing creating a blindspot for us during our climb. I also pointed out to my student how on our MFD (Multi-Function Display); that Aircraft Y was the icon on the parallel runway with '00' above their marker. Their marker was highlighted yellow to caution us due to how close they were. At the time I was pointing out Aircraft Y's icon on the MFD to my student; we saw their icon start turning toward our location. I immediately took controls and pushed the nose down to start a descent. Their icon brushed over our icon on the MFD and showed '+01;' indicating that they were 100ft above us. During this incident; I could never see the aircraft due to the blind spots on our aircraft and I would not have had the situational awareness for where they were if it weren't for the traffic display on the MFD on our aircraft. ATC audio available was grumbly and cut off after Aircraft Y's initial reaction to Tower; but I remember immediately querying Tower whether or not Aircraft Y was supposed to be following us. They denied it and told us that we were supposed to be following them. There was no instruction for us to follow that traffic during anytime of this incident. I replied along the lines of that they had just passed over and we almost had a mid-air with them. The Tower later instructed us a few seconds after my query to make left closed traffic. Adrenaline started kicking in for me and I requested a full-stop terminate because I was not fit to fly anymore from this point. I grounded myself for the rest of the day due to how close this encounter was and the anxiety I had from this incident.