Thanks @carolhmj for looking into this!
After a long detour and a deep dive into *GL extensions (to better understand how Basis Universal compressed GPU textures are handled in different environments), I took a fresh look at my KHR_parallel_shader_compile issue today, and at your answer.
Confusing nomenclature indeed! 
What you write makes perfect sense though, and the presence of a “WebGL * Extensions” section right next to each “WebGL * Driver Extensions” section should have been a red flag.
In fact, the extensions listed under “WebGL 2 Driver Extensions” are the same returned by gl.getSupportedExtensions() (with the exception of MOZ_debug). And in my environment, KHR_parallel_shader_compile is not on the list.
Chrome on the other hand, seems to only list in chrome://gpu/ the extensions supported by the driver. As I could find GL_KHR_parallel_shader_compile listed there, I assumed wrongly that if the browser reports (driver) support for GL_KHR_parallel_shader_compile, that is sufficient to guarantee browser support.
And to add to the confusion… Caniuse reports that the extension should be available in Firefox since v80. But guess what? This info is incorrect, and there is an open issue about it!
Phew… 