So, the task in question is "[Fathers day] Finish 25 mission of type Airport Missions (Specialization)". I think the problem is that there are no missions that have the tag "Airpot Mission (Specialization)", there is only the general "Airport Missions", but that apparently does not count for the task.
Here is what I've done on my end that led me to the conclusion:
I assigned a single Firestation with the Airport Specialization to a separate Dispatch Center ("Farm DC"). Every other Dispatch Center was deactivated so only the Farm DC spawned missions. After getting some larger (and time intensive) plane fires/crashes I also activated the "Own Coverage Area" option for the Farm DC. In the end, I cleared a potpourri of: Landing Gear Failures, Braking Overheats, Large Aircraft Crashes, Debris on Runways, Burst Tires - Aircraft, Jetbridge Collapse, Cargo Plane Fires, Airport Terminal Fires, Aircraft Leaking Fuel and Engine Fires. A few of those were even spawned before I separated the Station over to the Farm DC. But all throughout, the task counter stayed at 0/25...