Here's information on the bugs I found when parallel mode is used:
1. When I use the option -g to latexmk to force all rules to be run at least once, the parallelization code vetos the running of the custom dependencies because the built file is "already up-to-date (built by concurrent process)". The code is using a test that is wrong, even though it appears natural.
2. More importantly, when I use parallel mode on my colleague's document that I mentioned in my previous e-mail, it keeps trying to rerun all the custom dependencies until it hits the limit of the maximum number of allowed runs. I conjecture from the behavior that there is something going wrong in the use of dependency information.
If you are interested, I can send MWEs. John

