A320 Captain reported engine fire detected after parking at the gate. Fire was quickly extinguished with no injuries.

Narrative

This event began with a routine landing in ZZZ followed by a taxi to the gate where we waited 5-10 minutes for the ground crew to park us for a remain overnight. I shut down both engines at the gate; did the shutdown checklist; and the FO (First Officer) and I leisurely gathered our belongings while we waited as the passengers deplaned. Eventually FO stated that they were going to do the post flight walk around inspection and left the flight deck. I had previously entered a logbook deferral sheet in the logbook that both engines required oil servicing.Up to this point everything was routine and there were no unusual indications or messages from the aircraft. A few minutes later the FO reappeared in the flight deck and stated I think the #2 engine is on fire." They then told me that looking at the engine from the rear it looked like there was a flickering flame inside the engine as you looked in the tailpipe. There were no fire indications or other anomalies in the flight deck. There were still about a dozen passengers on board so out of an abundance of caution I called the Tower; explained the situation; and asked for the fire trucks to be sent. I later learned that the crew chief had independently called for the fire trucks a minute or two earlier. I did not initiate either the engine fire or tailpipe fire procedures since it was unclear to me what the nature of the problem was at this point. I then went down to the ramp.As I approached the number 2 engine I observed that the ground crew was unloading baggage from the rear cargo compartment just a few feet behind the engine. I walked to the aft end of the engine with the FO and could clearly see the flickering light from a fire when I looked in the tailpipe. Moments later the fire trucks arrived. Just as the fire personnel walked over to us the fire apparently went out on its own. We observed for a few minutes and after discussing the situation with the FO; fire personnel; crew chief and AMT who was there to do the oil servicing we concluded that the immediate danger was over and any subsequent actions would be done by Maintenance. We did note that there was a clear trail of oil spots leading back from the engine to the taxiway; and a fresh puddle of oil under the engine. I called Maintenance Control to inform them of the situation and returned to the flight deck to enter the problem in the logbook and conduct the checklist.One of my challenges in this incident was in deciding how to accurately describe the problem. It was also late in the evening after a long day and I was certainly affected by fatigue. I wrote that the problem was a "Tailpipe Fire" and added some additional description. In hindsight this was probably wrong. I should have described it as a "Fire internal to the engine viewed through the tailpipe." Later at the hotel the chief pilot called and said that Maintenance Control was confused about what the exact problem was. Luckily the FO had taken pictures and video that I sent to chief pilot and they shared with Maintenance Control; which helped clarify things.I have thought about whether I should have initiated either the engine fire or tailpipe fire procedure. However; I never had any indications of an engine fire in the flight deck and almost as soon as I went down and observed it the fire went out with the fire department now on scene. Similarly I had no indications of a tailpipe fire. I subsequently researched in our manuals and have determined that there is no definition or description of "tailpipe fire" anywhere that I can find. I don't actually know what constitutes a "tailpipe fire" versus other types of engine fire; and when to apply which procedure after shutdown.Maintenance eventually determined that there was enough fire damage to the #2 engine that an engine change was required. Among other issues; one of the oil gaskets was found to be out of place and severely damaged. I would note that a previous crew had written up a suspected oil leak from the #2 engine earlier that day and it was signed off by Maintenance with no repairs made. Based on that earlier write-up; the oil consumption on our flight; and the visible oil leak after landing; I concluded that we were very lucky not to have had an engine fire in flight.First; Maintenance should be more through in investigating reported engine oil leaks. Second; it would be extremely beneficial to receive additional training on how to identify and distinguish post-shutdown engine and tailpipe fires."