Cannot launch Parallels Desktop after upgrading to macOS Catalina? After having installed macOS Catalina, I cannot launch Parallels Desktop anymore.

Launching Parallels Desktop from the dock results in the message in the image below. It says that I should "Allow" Parallels Desktop in "Security and Privacy" settings, but it's not showing up there.

How can I manually "Allow" Parallels Desktop in "Security and Privacy" settings, when it's not showing up there?
Your screen capture shows the Security and Privacy > :General" tab. Did you look under the "Privacy" tab?

There will be a list of categories on the left and then the permissions of each app on the right. You will need to select the categories on the left one at a time (I'd start with "Accessibility") and authorize any instance of Parallels that requests permissions for that app..

To do this you will first have to click the padlock icon and authenticate with an admin password.