You got it backwards. It's a jet plain with the sound of a propeller plain. Anyway, it is not a slip-up. It was done on purpose. IT'S A JOKE! It's a nonsense comedy full of inaccuracies.
Actually it's the other way around. The outside view shows a jet aircraft (a Boeing 707 or a shorter model of the 707, a 720), but it is accompanied by the sound of a prop-driven plane. I have always assumed this was supposed to be humorous. Also, look for them to refer to the plane by the wrong model number in the sequel, calling the original plane a 767 (if memory serves), which only has two engines. I'm picky, I know. But none of this takes away from the hilarity of the movies.
The directors used the prop plane sound as a joke because they were originally going to use a prop plane for the movie, but the production company told them to use a jet plane instead.
It's too bad so many people missed a lot of the humour! Airplane spoofed every movie and TV show that has had anything to do with aviation. Many TV shows in the 60's and 70's would show a DC-8 taking off, A B707 in flight, and we would get that great under-belly shot of the gear coming down-- B-52 gear! All supposed to be the plane the hero was taking to where ever.
Airplane was a spoof and a hoot from the opening with the tail of the plane sticking up through the clouds accompanied by the Jaws theme, through to the end! Sit back and enjoy it! What's the vector Victor?