These three words look similar but they couldn’t be more different in meaning, referring to three distinct regions in the world.
Persian is anything related to Persia, an ancient region with some of the oldest writing systems (the Cuneiform) and the most powerful empires (Achaemenid Empire). It was the Persians that the Greeks were always fighting against.
Most of ancient Persia is the modern country of Iran which got its name from the Aryans (‘Iran’ = land of Aryans).
Prussian is anything related to Prussia, a region in today’s northern Germany and Poland. Prussia was famous for its ‘Prussian Blue’, a dark blue pigment that was used for engineering drawings, art, textile and pretty much everything. It’s because of Prussian blue that the famous Prussian military uniforms were blue and that “Blueprints” are called blue prints.
By the way, the name Prussia also has an alternate Latin spelling – Borussia, the name for a famous German football team today!
Parisian is anything related to Paris, especially a person living in Paris, France. Enough said!