Pilot departing on an ATC assigned heading and altitude reported taking evasive action due to a NMAC with a departing aircraft from a parallel runway.

Narrative

ATC cleared Aircraft X; my airplane; for takeoff from Runway XXL and to climb to an altitude of 2;000 feet on a heading of 090. Around 700 ft. a turn was initiated to the 090 heading. We were told to contact departure. After switching frequency but before contacting departure control and approaching the departure end of of Runway XYL; I noticed that Aircraft Y was rotating and climbing very rapidly to our direction. I immediately turned right to a 150 heading while increasing the climb rate to avoid the rapidly climbing Aircraft Y. The Aircraft Y appeared to have turned left toward our direction which made their flight path pass virtually right beneath us around 100 ft. lower. I resumed the 090 heading and contacted departure to tell them what happened. He then asked 'what heading did they give you?' And I responded with '090 up to 2000''. He said ok and the flight progressed normally afterward. I believe this occurred due to ZZZ having West and East control towers. West Tower and East Tower may have had a communication breakdown.