A320 Captain reported the aircraft experienced a tail pipe fire after parking at the gate.

Narrative

Upon arrival into gate; shutdown/parking checklist/deplaning normal. After a few minutes; Maintenance came on the flight deck off-handedly mentioning that I had a tailpipe fire and Fire Department (FD) had been notified and was on the way. Passengers were almost 2/3 deplaned at that point; I looked out the window of the jet bridge to see Maintenance personnel standing directly behind the tail cone of eng 1. I instructed the FO (First Officer) to have his Tailpipe Fire QRH ready. As personnel were directly behind the engine; I did not have the QRH procedure run. Maintenance came up to the aircraft again; taking the fire extinguisher from the FWD FA (Flight Attendant) station and heading downstairs. I followed them down and looked in the tail cone as they were getting the extinguisher ready; there was a small; but steady drip of oil that was on fire as it fell from the tail cone into the rear bypass area. Maintenance sprayed the extinguisher in the tail cone; the fire extinguished immediately; and I returned to the flight deck. FD arrived; took my information; asked if there was any sign of fire on the flight deck and whether there were any indications out of the ordinary; to which there was not anything unusual noticed by either flight deck crew. The maintenance crew should have a fire extinguisher handy in each vehicle for occasions like these. The extra time wasted going down to the engine and coming back up for the aircraft's extinguisher wasn't critical this time; but could have averted disaster had anything more dire happened. I did not elect to begin an evacuation because most of the passengers were already off and I think it may have caused more problems than solutions in regards to safety. Also adding to the decision was the lax; off-handed nature of the Maintenance personnel; to include other Maintenance personnel standing directly behind the source of the fire.