For all intents and purposes Jet A is (virtually) the same as Diesel.
Avgas is virtually the same as high octane pump gas.
So no, you can't burn only Diesel in a petrol engine or the other way around.
You can run small amounts of contaminated fuel as in about 15-20% petrol in a diesel engine and vice versa.
Turbo prop and jet engines can pretty much run on anything that is combustible but it wouldn't be certified to do so and therefore un-airworthy.
The Russians used to be famous for having airplane engine that could run on the same shabby fuel as their tanks.
Great example is the Sukhoi 25 "FrogFoot".
In short, no you cannot run Diesel or Jet A in a airplane "petrol" engine