GA instructor pilot reported an NMAC with another aircraft in the pattern which did not follow ATC instructions.

Narrative

The following event took place during a training flight in the airport traffic pattern during a typical training flight practicing takeoffs and landings with a student seeking a Private Certificate. My student and I were on Runway 30L just lifted off and in our initial climb. Tower informed us to continue straight out and traffic was 12 O'Clock in the left crosswind; they were our traffic to follow for left traffic for 30L. My student and I were both scanning straight ahead for the traffic based on ATC Traffic advisory. We saw the traffic at our 1230 to 1 o clock position from our perspective; was the high wing Cessna; and we were flying a low wing Piper. My student immediately lowered the nose leveling off and stopping the climb to avoid any further decrease in separation. As instructor; I immediately informed Tower traffic was in sight and currently was passing directly over top of us. The high wing Cessna had departed the parallel runway; Runway 30R; on an extended upwind and just entering left traffic ahead of us; per ATC; but appeared to have continued their turn from upwind to downwind instead of flying a crosswind leg. Tower did inform the high wing Cessna after passing they had to widen their downwind because they turned into upwind traffic on the parallel. No other conflicts occurred. From my position in the low wing Piper; the high wing Cessna appeared to be about 400 feet above us when passing over top.As a result; I discussed with my student that even though we both were listening to the ATC Traffic advisory and scanning for that traffic; to not let that interrupt our situational awareness of traffic on the parallel runway. Despite this incident both aircraft being on same frequency; it is possible they can be on different frequencies. Both the student and instructor learned about the requirements of situational awareness; and to yes look for traffic ATC points out; but keep that scan going being careful not to expect to see that traffic cross in front of us. While we were still on our straight out instruction from ATC initially; situational awareness is key with parallel runways; especially at airports where it is common for ATC to assign that aircraft left traffic for that runway even though it doesn't follow published pattern directions. It's still possible for traffic to cross ahead of or behind you in the crosswind legs. My student and I were able to take the appropriate actions to avoid any further loss of separation and avoid a collision after visually acquiring the traffic.