B350 Captain reported a traffic conflict while departing from a parallel runway as ATC had issued a clearance that placed the reporter's aircraft in close proximity with another. ATC had assumed the reporter was going to turn to a left heading and failed to provide safe clearance between the two aircraft that were taking off from parallel runways.

Narrative

On Day 0; I was lifting off of Runway XXL at ZZZ. My initial clearance was; 'On departure; make a right turn heading 360; radar vectors direct to ZZZZZ; then as filed. Climb and maintain 2000 ft.'I had climbed to about 600 - 700 ft. before I was to make my right turn. Just before I was to initiate my turn to the right; I received a TCAS alert. The multi-function display showed the traffic concern as being 300 ft. above me and very close. I quickly looked up to scan for traffic as I was continuing to climb and almost immediately spotted the aircraft ahead of me within close proximity of my aircraft; and much closer than I might normally be comfortable with; especially when it comes to simultaneous operations with parallel runways. I then quickly and somewhat abruptly made a sharp right turn; banking as much as 45 - 50 degrees to the right to evade any potential conflict with the traffic ahead. The aircraft ahead of me was a Piper Cherokee. I had estimated that I had come as close as 300 - 400 ft. of that aircraft before going to my initial heading of 360 degrees. I proactively advised Tower that 'I am turning inside of that aircraft ahead' thinking that they might have received an alert in the Control Tower. There was no response from Tower. Once I was headed in the right direction of 360 degrees; and nearing my level-off at 2000 ft.; I queried Tower about the traffic conflict. They responded back; 'Sorry about that; that was my fault. I thought you were going to turn to a left heading of 150 degrees.' After a couple of minutes; I was handed off to Approach and the rest of the flight was uneventfulLater that evening; I initiated contact with Tower to speak with the Manager about this incident. They were not working that evening; but I was advised that my message would be relayed to them. I called again on Day 1 afternoon to see if I could speak with the Manager and they were still not in; but I was told that they got the message and that they would be calling me within 48 hours. On Day 2; I received a call from the Tower Manager. We spoke about the incident and the concerns I had of it ending in a catastrophic accident versus just a near miss. They were very apologetic and professional in relaying to me what they saw as to a few possible reasons why this might have occurred in the first place; yet not excusing the mistake that was made by the Tower Controller at the time that I was flying. They had reviewed the audio recordings and looked at the radar information which showed the near miss. They also checked with a system that is used by aircraft control towers. Through the use of that technology; they were able to ascertain that my aircraft; King Air 350; and the Piper Cherokee were as close as 100 ft. of each other as I was banking away in my abrupt turn to the right. The Tower Manager let me know that this incident was being investigated and analyzed in order to try to prevent something like this from happening again in the future. They also assured me that the responsibility or blame for this incident should be falling squarely on Air Traffic Control because I correctly read back and executed my clearance as it was laid out.I believe that the cause of this event might have been related to the workload of the Tower Controller; along with a failure to recognize or be aware of the potential traffic conflict that was unfolding in those couple of minutes when both aircraft were departing their respective runways. The other cause of this event was the self-admitted assumption or belief from the Air Traffic Controller that I was supposedly going to turn left initially and not right. There might have been a breakdown of communication from Ground Control to Tower Control of what my specific clearance was; and not what was assumed. The factors selected below are in reference to my impression of what was probably happening in the Control Tower at the time.I am of the opinion that if there are simultaneous pattern operations; then any IFR clearances should have an initial turn away from the parallel runway. In this particular case; if ATC wanted me to take an initial heading of 360 degrees from Runway XX Left; then it should have been a left turn; from Runway XXL; all the way around to a heading of 360 degrees; almost resembling a left teardrop over the field. By the time most aircraft are back overhead the field; they would most likely be significantly above the traffic pattern altitude for most of the light aircraft that might be doing right-closed traffic patterns for Runway XX Right. They typically will fly their patterns between 1000 - 1100 ft. AGL. The other thing I would recommend is for tower controllers to be a bit more proactive about any training aircraft which might be experiencing any drifting on the upwind leg; which might conflict with other aircraft departing from and climbing up from the parallel runway.