Hispano-Suiza has delivered the world's first target-door reverser for a big turbofan, the A320's CFM56
The extending petal-shaped doors type is a Hispano-Suiza specialty. It's a French company and one of the three major suppliers of reverse systems.
In the 80's Hispano-Suiza had to be innovative to capture a bigger market share. Their door type was the first ever on a high bypass engine. They believed investing in R&D will help increase their sales (it did). The petal reversers also double as airbrakes.
Why not on the Boeing 737?
Since the CFM56 on the 737 is close to the ground, the doors would scrape the pavement. And—more importantly for other aircraft types—Hispano-Suiza's competitors are Rohr and Boeing. Boeing definitely would like to keep things in-house with their translating/cascading reversers.
Nowadays with the jet engines of the smallest regional jets to the wide-bodies being as wide as the fuselages, the cascading type is the better choice.