A320 pilot reported after shutting the engines down at the gate; a ramp worker informed the flight crew that engine 1 had a tailpipe fire. Airport responders put the fire out and passengers deplaned normally.

Narrative

Upon arrival into ZZZ at Gate XX with the APU on MEL; we shut down engine 2 and then waited for the ground power to be connected. After ground power was connected and engine 1 shutdown; we were doing our shutdown checklist when a ramp worker knocked on the cockpit door and informed us that we had a tailpipe fire in engine 1. We immediately performed Engine Tailpipe Fire QRC. We had no indications in the cockpit that there was a tailpipe fire. Since the APU was on MEL we were not able to dry crank the engine. Without knowing if the fire was still there or not I made the decision to call ARFF (Airport Rescue and Firefighting). When they arrived; they shot water up the tailpipe. The passengers were halfway through disembarking and felt that an evacuation was not necessary as everyone was getting off in a timely matter. I looked out at the engine and did not see any flames before or after the ARFF had put water on the engine. Having the APU on MEL; it prevented us from dry cranking the engine which most likely would have prevented us from calling ARFF and having them to put water on the engine. The APU was on MEL for 7 days and if this was fixed it would have us able to complete the Engine Fire Tailpipe QRC.