I bet there's a hole at the top!
Parachute designers over the decades have found a need for a small hole in the top to allow smooth airflow. Leonardo didn't include one, but I bet the Swiss one has one. Also, when you compare leonard's sketch and the actual parachute, you'll note a dramatic difference in the ratio of parachuite line length to parachute width. This is what enables the design to do away with the wooden frame.
In a sense, it's unfair to say that this is "Leonardo's design". In fact, it's a novely modern parachute designed to mimic Leonardo's design...