The Captain is sitting in the right seat in the cockpit, and there seems to be no first officer, which would be required on an aircraft that size.
The exterior shots are of a BAe 146, but the interior of the plane is of a much wider-body aircraft.
Passenger aircraft do not carry parachutes.
Father Ted is feeling nervous on the airplane, so he places a mobile phone call to Larry Duff. Larry doesn't answer, but the FCC banned cell phone use on airplanes in 1991 while the show was broadcast in 1996.
The Blind Priest says he's laughing because he's listening to Mr. Bean on the Airline Radio headset and not because he's throwing newspaper at the back of Father Ted's head. "Mr. Bean" is visual comedy, the character rarely says anything. This is almost certainly intended as a joke.
When the first priest stands up to share his reasoning for why he deserves the parachute, a boom mic is visible.
During Father Cave's speech, the boom mic is visible.
In the first class bathroom, the boom mic is visible.