A Piper Arrow pilot reported NMAC with opposite direction traffic climbing towards them.

Narrative

While flying VFR northbound at 6;500 feet MSL approximately 25 NM northwest of ZZZ; I was in contact with ZZZ Approach with VFR flight following; enroute direct to the ZZZ1 area for sightseeing. ATC advised of traffic departing ZZZ2 several miles to my northwest and instructed me to turn 20° right. At this time the traffic display showed the aircraft more than 1;000 feet below and several miles away; and I continued to visually search for this traffic.Shortly afterward; ATC issued an urgent instruction to conduct an 'immediate climbing right turn.' I complied with a full throttle climbing right turn. Although I did not have the traffic in sight at that time; I immediately followed ATC's instructions; which may have prevented a collision. During the maneuver; the other aircraft continued a rapid climb; further reducing vertical separation even as I climbed at more than 1;000 feet per minute. According to my traffic display; the vertical separation at the closest point was less than 100 feet; and our aircraft appeared nearly co-located on the display. I still did not have the aircraft in sight; as I believe it was behind and below me at the time. After conducting this climbing turn for about 90°; ATC advised that the other aircraft had me in sight; I then visually acquired it as it passed very close; parallel to me; in a left turn and descent. When I asked ATC about what had occurred; I was told the aircraft had been in communication with controllers on a separate frequency.Following the near miss; I assessed whether I felt comfortable continuing the flight and decided to proceed. Nearly an hour later; however; I realized I had been mentally distracted by the earlier incident. As a result; I inadvertently started to enter an area of active parachute operations without recognizing it in advance. Upon seeing a maneuvering aircraft ahead of me on my traffic display; which I realized was a descending jump aircraft; I immediately exited the area.I am submitting this report to document the near-midair collision risk posed by the other aircraft's rapid climb and frequency separation; and to highlight how the event impacted my subsequent situational awareness for the remainder of the flight.