Since it is working (somewhat) when it is in park, I have a feeling the problem may be in the lamp position module for the left side.
I attached the wiring schematics for the exterior lighting system. I highlighted the wires related to both the left and right parking lights. Additionally, the last two pics show the module I am referring to. As far as the delay, it can't be a ground issue because the brake lights use the same ground location and they work properly. So, I suspect it could be the module. Also, the j519 module (electronic control module) may be the issue.
If you haven't already, you may want to consider performing a scan of the can-bus system. CAN stands for controller area network. Basically, all the modules/computers are tied together via a few wires. If there is a failure in any of the modules, this procedure should identify it. Here is a quick video showing how it's done. Note: When this is done, we need to pay attention to the J519 control module for codes.
https://youtu.be/InIlnsjOVFA
If possible, have that done. Hopefully, it will identify a code that is related.
Take a look through the schematics below. I had to cut them in half to make them readable for you. I did, however, overlap them so you can follow from one to the next.
Let me know what you find or if you have other questions.
Take care,
Joe
See pics below.
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Thursday, April 29th, 2021 AT 8:16 PM