PA28 Pilot reported failure to correct for wind on climbout resulted in a NMAC with departure aircraft on parallel runway.

Narrative

It was my first time flying out of ZZZ; and my second time flying in this particular make and model PA28; as most of my flight training was done in a Cessna 172. I was learning this make and model as this was a training flight and a flight where the engine was being broken in as well. Startup and taxi were normal as we proceeded to [Runway] XXR. After completing our run-up; we requested a straight-out departure from XXR. On departure; I did not apply enough wind correction; nor did I strictly maintain the straight-out heading; which caused me to drift left toward the parallel Runway XXL's approach end. We crossed just behind the touch-and-go aircraft which departed off of XXL. We had the vertical separation of approximately 200 feet and a horizontal separation of approximately 100-150 feet; as best I can recall. We spotted the aircraft from the right window and immediately climbed and corrected our heading. The flight continued without any further incident. There are several areas where I could improve and correct for future. Firstly; I failed to acknowledge and understand the potential risks posed by traffic on the parallel runway and did not pay attention to it. I should have properly briefed the dangers related to the parallel runway and will pay close attention to it every time from now on. Secondly; I did not account for the slight wind from the right that we had and should have applied proper wind correction to avoid drifting to the left. I should also have maintained a straight-out heading strictly to avoid the aircraft from drifting toward the parallel runway's approach end. I should have been more prepared and extra careful considering the fact I was very new in this plane despite flying with my CFI. Unfamiliarity and not enough rudder inputs along with proper wind correction was a significant factor. Going forward; I will pay closer attention to and acknowledge the risks associated with parallel runways. I was unfamiliar with this aircraft and learning it which is why I should've been even more careful. I will use wind correction inputs more effectively to maintain the departure heading and avoid drifting onto parallel runways. I will use this event as a learning point in my career and make necessary changes to my pilotage in order to prevent similar situations. This incident will make me a more diligent and safety-conscious pilot. I am committed to improving my decision-making and ensuring that situations like never happen again in the future. Thank you for your time.