I think your problem is that little tab I circled in the pic from your video. It's supposed to catch the bottom side of the rotating section to cause it to "shift" into park.
Try raising it up if possible. It may need to be replaced (if that's possible). But it appears that corrosion has worn it down.
In the 2nd pic, you can see how the tab engages the transmission when it's turning counter-clockwise. That must happen in order for the wipers to park.
HTH.
Doug
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