GA flight instructor with student practicing touch and goes at PDK airport reported an NMAC requiring evasive action to avoid a possible collision.

Narrative

I was performing a traffic pattern training flight with a student. Runways 21L and 21R at PDK were active; and we were assigned right closed traffic for Runway 21R. Throughout the flight; there were various amounts of aircraft in the traffic pattern with us. The first 3 laps in pattern went as expected with no issues. After the third touch and go on the departure leg; Tower instructed us to perform an early crosswind turn. At PDK; this is not an unusual instruction to receive while flying in the pattern due to the simultaneous parallel departures which occur quite often. This particular command to turn an early crosswind seemed unusual to me though; as there was no traffic observed departing the parallel runway. As instructed; we began our crosswind turn at approximately 500 AGL. We continued our crosswind leg until 1 mile laterally from the runway; then began the turn into downwind. After becoming established on the downwind leg; we received an ADS-B traffic advisory for an approaching aircraft at approximately 3'oclock and at the same altitude as our aircraft. With no immediate visual contact; I asked the Tower over frequency if they could see any aircraft near our position on radar or knew of any conflicting traffic. I received no response and briefly after; the controller began speaking to another aircraft on frequency. The traffic advisory informed us that the aircraft was getting closer. Since we still had no visual contact with the aircraft and Tower was not responding; we slowed the aircraft down and gained visual contact with the traffic; which was approximately 10 feet below us and 25 feet to the right of us; flying parallel on the downwind at approximately the same speed. This was caused due to the traffic turning a significantly early crosswind and downwind without having our aircraft in sight; who they were assigned to follow. Since we were to the left of the traffic; we deviated slightly to the left and increased speed to distance ourselves from the aircraft. I made a call on radios to the Tower advising of the near miss and requesting a full-stop landing clearance. The traffic slowed down to help separation and we proceeded to fly a normal traffic pattern to land. There were a few contributing factors to this near miss including the Tower's negligence of aircraft location in the pattern; the inconsistency in distances of traffic patterns flown; and the reduced visibility due to low/high wing aircraft. Personally; I; along with many other pilots at PDK; believe that this air traffic control Tower should not be used for controller training. There is a high volume of flight training at PDK which is why there should only be highly-skilled air traffic controllers helping to maintain traffic separation between low-time pilots. Oftentimes; there can be 4-5 aircraft in the traffic pattern for one runway at PDK which is a very high workload for air traffic controllers and should not be handled by a controller with little to no experience. The other aircraft was also at fault; having flown an incorrect traffic pattern. According to the Airplane Flying Handbook and Airman Information Manual; the crosswind turn should begin when the aircraft is within 300 feet of pattern altitude and past the departure end of the runway. After watching the ADS-B replay of their flight; they began their crosswind turn almost 500 feet below traffic pattern altitude. Both of these factors contributed to this near mid air collision.