The post you refer to is only comparing different parallelization options, it doesn't explain how to enable it. You can refer to this post which explains how to enable parallel tuning. Optimizations and Parallel Processing • tune.
~That article tells the user how to register cores for parallel processing. It does not enable it, as far as I can tell. This is because you need to set the option parallel_over which is set to NULL by default. The question is where is this option set as it does not seem to be used from options(). I do not think this resolves the question, unfortunately.~
Edit: it comes from the control_grid()function and other control_*() in {tune}