How can I get around this? This is super frustrating. And it isn't just for Parallels. I get it for other apps too.
I had the same problem. This answer on a different copy of this question is a solution with a tradeoff: https://apple.stackexchange.com/a/327816/292890
Summary is: cmd-R on boot to get Recovery Mode, then open Terminal by clicking on Utilities > Terminal. Once open, enter the following command:
spctl kext-consent disable
It disables kernel extension security confirmations altogether, which of course hurts your security, but it does prevent all the confirmation popups. That's also less drastic I think than disabling SIP entirely.
I'm glad I migrated my home folder and applications to a separate partition, because it turned out that not even reinstalling the OS solved this problem unless it was a clean install.
I had this problem in High Sierra, updated to Mojave but it persisted. Reinstalled the OS, but no change. Finally wiped the partition (or APFS volume in this case) and reinstalled Mojave on it. Linked in my home folder, and the problem was finally gone. The same should probably work for High Sierra.