Small Aircraft; Low Wing; 1 Eng; Fixed Gear
Flight Crew Multiengine; Flight Crew Flight Instructor; Flight Crew Instrument; Flight Crew Commercial
Flight Crew Last 90 Days 200; Flight Crew Total 1100; Flight Crew Type 300
Communication Breakdown; Situational Awareness
ATC Issue All Types; Conflict NMAC
Flight Crew Took Evasive Action
Human Factors; Airport
I was cleared to takeoff from Runway 25L and was initially asked to fly straight out. Upon reaching 400 ft tower said right turn northwest bound approved and said monitor north tower. South tower didn't tell me there was departing traffic from the parallel runway westbound. I saw the Cessna as we were turning and adjusted our climb rate and direction to avoid traffic. The traffic crossed underneath us. Seemed very close to us; approximately 100-200 ft. TIS (Traffic Information System) traffic was unavailable in the aircraft at the time.
GA flight instructor reported ATC issued a turn towards traffic departing from the parallel runway at DVT resulting in a near midair collision.