the swinging bridge you are talking about still exists. It is in the swope park zoo.
I'm searching for information on an old bridge in Swope Park. I believe it spans a part of the Blue River. In the 1940's it was known as a "swinging bridge" and was part of the trail used to walk from the end of the streetcar line to the lagoon and swimming pool in the park.
I see nothing that sounds like it in this Historic Bridges of the Midwest website. Would appreciate any information on it. As I recall, it was only about 10 to 12 feet wide, steel construction with wooden deck. Warnings were posted not to swing it...but when five or six people "marched" on it in step with one another, it did swing.