A TRACON Controller reported an aircraft with one engine shut down descended into confliction with parallel runway traffic. The Controller did not realize their trainee had issued the wrong altitude to the aircraft.

Narrative

Was Training on the Radar 2 position. [Priority] Aircraft X was inbound via the ARTCC; so plenty of time for planning. We took radar on the [priority] who was tagged up for Runway XXC which was the pilot's request. I should've asked sooner than I had; but around the ZZZ1 airport I noticed the aircraft had not descended out of 9500 ft. yet which would be considered WAY high. I asked the Supervisor who was running the wall 'what is the nature of this [priority]'; which is when I was informed the aircraft had lost an engine. So at that moment I knew he was going to be slow to descend; fast and make wide turns. I instructed my Trainee to stop the [priority] at 4000 ft. and he said he had; then I told him I was going to 'talk' and to just listen. However I didn't verify his confirmation of stopping the aircraft at 4000 ft.I turned to the Radar 3 Controller and instructed them I needed them down with Aircraft Y to 3000 ft. because I was going through the localizer because the [priority] was high. They said they had done so; but he wasn't descending. I reiterated the need for her to push the aircraft down to 3000 ft. Finally on the third time she told the pilot to get down. By this time I had turned Aircraft X to join the localizer from the opposing base which is when I noticed he was descending out of 4000 ft. to 3000 ft. Radar 3 pointed out the traffic to Aircraft Y and eventually got the visual separation; however I believe separation would have been lost by then. The closest I found via FALCON replay was 1.69 Miles and 200 ft. No evasive action was taken on my part. I found it; while not the required spacing; to be the best course and safest course of action to let Aircraft X continue as they were in the turn. Both planes landed without incident. I would recommend if not already a policy that the In charge person coordinate an [priority] with a lost engine to be a more advantageous position to descend and get in safely; and to make sure whoever is working the airplane is made aware of the nature of the [priority]. Essential be proactive; not passive as they were just sitting watching it happen. I take full responsibility for not verifying that the [priority] aircraft was stopped at 4000 ft. and took my trainee at his word. Operational oversight was definitely an issue in this situation. I felt I was basically running the entire wall; training; running my position; the other final and the CIC (Controller in Charge) doing on the spot corrections. The Supervisor could've definitely played a more aggressive roll in the situation; however I own that there were a few missteps by myself.