I recently started having this issue on a 2002 Chevrolet Impala, where the wipers get stuck in an upright position after wiping and won't go into the parked position when turned off.
It appears to be a common problem especially on many GM vehicles. The frequently proposed answer is to bend a tab back into position near the wiper transmission: https://www.wikihow.com/Fix-Windshield-Wipers-Stuck-in-Upright-Position
However this tab did not appear bent in my case. In case it helps others, I decided to post this question, and the solution I found, in the answer below.