Hide Parallels's VM icon from the dock
It's pretty ridiculous that this slipped both me and Parallels's support to the point that I went through all the effort above, but there it is - if you go to Parallels's preferences, this setting is under General > Virtual Machine icon. Select none.

I believe my question has implications extraneous to this particular case, but since the submission is under the Parallels's title and confers much specificity to the case, I'll close it as it is.