A Tower Controller reported a flight school aircraft on short final reported a NMAC with another flight school aircraft that had deviated from their instruction to land on the parallel runway.

Narrative

Runways XXL/R in use; VFR conditions; multiple aircraft not responding to transmissions and not using call signs consistently. Aircraft X was operating left closed traffic behind Aircraft Y Runway XXL. Aircraft Y remained left closed. On the upwind Aircraft X was instructed to extend upwind; make right closed traffic; change to Runway XXR for 3 CAT III aircraft coming inbound Runway XXL. Aircraft X delayed read back was advised that their subsequent read back was garbled and unreadable; and that the call sign was not used. The pilot replied with somewhat unusual phraseology referencing 'last command'. Aircraft X was told 2 additional times that they needed to use call sign with transmissions and finally that it is required. Aircraft X then acknowledged right down wind assignment and changed to Runway XXR. The pilot made a very close traffic pattern to which I made a comment to my Ground Controller. I did not advise Aircraft X of this because I felt with the communication issues; it would further confuse the pilot and I had other tasks to complete. Aircraft Y was cleared for the option; unable to stop and go; behind traffic on final and inside of traffic 9 mile final. Aircraft X was cleared to land Runway XXR; traffic issued for multiple aircraft landing XXL. Pilot read back correctly and said traffic was in sight. After exiting traffic at taxiway XX; Aircraft Y on short final advised that they had a 'near mid air with Aircraft X' and that they were going around/turning to avoid. At that point; Aircraft X was observed short final XXL in very close proximity to Aircraft Y. Then Aircraft X made a sharp right turn low level to land Runway XXR. I did not transmit to Aircraft X during this time. Aircraft Y avoided traffic; seemed a little shaken on frequency; I told them to fly abeam Runway XXL; left side; and I called their crosswind to be re-sequenced. Aircraft Y landed without further incident. Aircraft X was issued the brasher warning. I would have scanned the approach corridor more. I felt that due to the excessive transmissions; my attention was delayed and I wish I had re scanned to have seen the situation as it developed and made corrective action. Also both aircraft involved were flight school aircraft based that fly every day here so I also had expectation bias that Aircraft X was used to common practices of using both runways.