PA-28 Cherokee/Archer/Dakota/Pillan/Warrior
Flight Crew Private
Flight Crew Last 90 Days 35; Flight Crew Total 220; Flight Crew Type 216
Situational Awareness
ATC Issue All Types; Conflict NMAC; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance
Flight Crew Took Evasive Action
Procedure; Human Factors
Contacted ZZZ Tower and requested to track the LOC XXR approach inbound with the intention to circle to land on Runway XYL. We were instructed to proceed as requested and report a 5 mile final for Runway XXR. Upon reporting our 5 mile final we were instructed to circle east and enter a left downwind for Runway XYL. During our circling maneuver Tower informed us that we would be number 2 for the Runway following traffic inbound on the VOR XYL approach and to advise traffic in sight; and Tower asked intentions after landing to which we replied we would like to depart the pattern north east to ZZZ1 airport. Tower acknowledged and told us to expect a right turn after landing back over to ZZZ1. We spotted the traffic and were cleared for the option number two following the traffic on final on XYL. We turned final and the traffic ahead was high and went missed following their approach so we landed on Runway XYL while a P3 was holding short XYL. After landing we conducted a touch and go and was climbing upwind with traffic on the parallel Runway XYR in sight. We then contacted Tower and confirmed that he wanted us in a right turn to depart the area. Tower then said to reference the traffic now turning crosswind ahead and that our right turn was approved with traffic in sight. We began our right turn northeast and the other traffic on the parallel appeared to turn right downwind for XYR. We then noticed there may be a confliction if we continued climbing so we leveled off at 500 ft. to pass below the conflicting traffic. Tower then informed the downwind traffic that we had them in sight and we would pass below to which the conflicting traffic reported us in sight as well. We remained at 500 ft. until clear of conflict and then continued our transition and climb over to ZZZ1.
PA28 Pilot reported NMAC during departure climb from airport.