ZZZ TRACON Controller in charge reported a pilot reported conflict traffic that overshot descent clearance altitude resulted in a NMAC.

Narrative

On Day 3 I was the Standalone CIC (Controller in Charge) and received a phone call from the pilot of Aircraft X. The pilot informed me he was calling to report a NMAC that occurred in the vicinity of ZZZ airport on the evening of Day 0 around XA:30. The pilot was filling out a report of the incident and followed up with our facility while doing so. The pilot of Aircraft X had stated that he was issued traffic by the controller on Aircraft Y; which is a jump aircraft that operates at both ZZZ and ZZZ1 airports. The pilot of Aircraft X stated that they could not see the aircraft visually but observed it on their traffic display. Aircraft X stated that the controller instructed Aircraft Y to stop their decent but it sounded like Aircraft Y had missed the control instruction to stop their descent. Aircraft X informed me that they received and responded to a TA from their aircraft and descended rapidly to avoid [a] collision. When they finally saw Aircraft Y the pilot reported to me that Aircraft Y did not have any of their NAV/RECOG lights on as they should have because they were operating at night releasing jumpers at night. Aircraft X reported that it wasn't until they told the controller that the other aircraft did not have any of their lights on that Aircraft Y then turned on what appeared to be a dim landing light. Suggestions: Continue to work locally to get company to enter into an LOA to standardize their operations.Involve FSDO AND NTSB to investigate this incident and the operations that are occurring at both ZZZ1 and ZZZ. This is another example of 'not if but when'. We are plagued by jump operations in Class E airspace operating in an unsafe manner thinking they are above the rules of civil aviation because they are operating flights releasing US military service members. We are on borrowed time with these operations; something needs to change before a loss of life occurs.