… and those sources would be wrong. As you surmised, Chrysler didn’t introduce their push button trans controls until '56. All of their automatic trans cars used that dash-mounted lever in '55, and I well remember it on our '55 Belvedere.
I’m not sure if this was the first year that they introduced that feature, but my father’s '63 Plymouth had a lever next to the trans push buttons that engaged the Park Pawl on the transmission.
Here is a pic:
Just like your '56 DeSoto, our '59 Plymouth lacked that feature, but whether Chrysler added it in '63–or possibly in '62–I’m not sure.