PSE 2021 will not load Luminar NEO, getting a 3rd party plug in error on startup

The Photoshop plug-in is not automatically installed into Elements.  Search your computer for the plug-in file LuminarNeo.8bf and copy it (in Windows) to the following directory:

C:\Program Files\Adobe\Photoshop Elements 2021\Plug-Ins

The plug-in should then show up in Elements under the Filter>Skylum Software menu item:

If you have not done so already, make sure you have the latest update from Luminar as it fixed some problems I was having using the software as a plug-in.  (I actually uninstalled Neo and reinstalled the latest version to fix the problem.)

You may want to contact Skylum support:

https://skylum.com/support

You posted in the PREMIERE Elements forum... do you mean PHOTOSHOP Elements?