Flight Instructor reported a NMAC after the Tower instructed them to turn crosswind into conflict with an aircraft departing the parallel runway.

Narrative

My student and I were conducting touch-and-go practice in the traffic pattern at IWA. We were flying left traffic for runway 12L. After completing our final touch-and-go; we requested a full-stop landing from the Tower. The Tower instructed us to 'fly straight out.' Simultaneously Aircraft Y was departing from the parallel runway; 12C. I wasn't initially aware of this aircraft. They were behind us and higher; effectively in our blind spot. At approximately 2;500 feet MSL on the upwind of 12L; the Tower instructed; 'Aircraft X; make right traffic for 12R.' Before initiating the turn; I checked the MFD (Multi-function Flight Display) for traffic; but due to the close proximity of Aircraft Y; I mistakenly believed it was an ADS-B 'shadow' of our own aircraft.As we turned right; we ended up within 200 feet of Aircraft Y; which was climbing out from runway 12C. The Tower immediately instructed us to 'continue on the upwind' after realizing the close proximity of the two aircraft. A few seconds later; they directed us to resume right traffic. After we turned back to the upwind; Aircraft Y adjusted their course to the south to avoid further conflict. Eventually I took controls and descended the airplane to break off. ATC did not issue to us a warning or mentioned that we were at fault. We landed safely back at IWA.