Tower Controller reported an NMAC resulted when an aircraft flying closed traffic on a parallel runway cut off another aircraft.

Narrative

I was working the Local 2 Position; VFR pattern in right closed traffic for Runway XXR. Aircraft X was handed off to me upwind from the parallel runway (Runway XXL). When the aircraft was on the upwind; I pointed out the traffic that he was going to follow in the pattern; a PA28; Aircraft Y. Aircraft X reported the traffic to follow in sight and proceeded to make right closed traffic as instructed. When Aircraft X got onto the downwind; I instructed; 'Aircraft X; follow the Cherokee ahead in the downwind; Runway XX right; cleared to land.' Aircraft X responded that he would follow the Cherokee and acknowledged the landing clearance for Runway XXR. After this I picked up another C172 (Aircraft Z) that was handed off from the parallel Runway (XXL) and I began to set his sequence to follow as he was south of the airport on upwind; I pointed out traffic that was in the upwind leg off of my Runway (XXR). It took a couple transmissions to get Aircraft Z to get his traffic in sight and to follow it. While completing this task my attention was visually on both of these aircraft south of the airport.Immediately after this sequence was set I turned my attention back to the north and simultaneously; I observed Aircraft X and Aircraft Y in close proximity and Aircraft Y keyed up and stated he was initiating a go-around and that he had just been 'cut off' by Aircraft X. He also stated that he would like to report a near midair.I told Aircraft Y to go around right side and extend upwind where I subsequently sequenced him with other downwind traffic and Aircraft Y continued operations in the VFR Traffic Pattern.This pilot had reported his traffic to follow 'in sight'. I'm not sure what we can do besides making our airspace a more positively controlled environment such as a Class C where everyone is separated by radar minima.This would of course change the entire dynamic of VFR flight at ZZZ. Free VFR flight as per 14 CFR is supposed to be see and be seen and to comply with any ATC instructions. This particular pilot did neither of those.To change FAA directives; ATC Procedures; or Airspace for one pilots error to me is unfair to the rest of the VFR and GA pilot community.The only suggestion I would have is to address the individual behavior instead of punishing all.