I tried to remove everything Parallels-related from my system (10.14.6). Found stuff in:
~
~/Library/Parallels
~/LibraryPreferences
~/Library/GroupSpaceContainers,
~/Library/Caches
/Library
/Library/Preferences
/var/root/Library/Preferences
/var/run
Then tried a new installation. It failed with the usual message "Unable to start Parallels services".
Below is the installation log, where I marked in blue two messages about a previous installation (so apparently I didn't remove all of it), and in red an error message about incorrect permissions.
2020-12-05 19:49:04+01 vBook pdfm-bootstrap[6526]: inittool[6520]: Disk space check: source size: 391332 Kb (586998 Kb will require)
2020-12-05 19:49:04+01 vBook pdfm-bootstrap[6532]: inittool[6520]: Disk space check: 1240834100 Kb is available
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6553]: inittool[6539]: Disk space check: source size: 391332 Kb (586998 Kb will require)
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6559]: inittool[6539]: Disk space check: 1240834052 Kb is available
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6561]: inittool[6539]: Found users in the system:
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6563]: inittool[6539]: * <...> /Users/<...>
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6564]: inittool[6539]: * Guest /Users/Guest
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6566]: inittool[6539]: Starting Parallels Desktop installation
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6572]: inittool[6539]: Bundle '/Volumes/Parallels Desktop 16/Parallels Desktop.app', version '16.1.1-49141'
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6573]: inittool[6539]: Check for previous version and uninstall if need
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6574]: inittool[6539]: Stopping Parallels Services from previous installation...
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6577]: inittool[6539]: Running '/Applications/Parallels Desktop.app/Contents/MacOS/Parallels Service' in stop mode...
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6596]: inittool[6539]: Running uninstaller
2020-12-05 19:49:12+01 vBook pdfm-bootstrap[6597]: inittool[6539]: Searching for bundles com.parallels.desktop.console and com.parallels.desktop.appstore
2020-12-05 19:49:16+01 vBook pdfm-bootstrap[6627]: inittool[6539]: Removing PUFS from previous installation
2020-12-05 19:49:16+01 vBook pdfm-bootstrap[6644]: inittool[6539]: Copying product into '/Applications/Parallels Desktop.app'
2020-12-05 19:49:19+01 vBook pdfm-bootstrap[6665]: inittool[6657]: Found users in the system:
2020-12-05 19:49:19+01 vBook pdfm-bootstrap[6667]: inittool[6657]: * <...> /Users/<...>
2020-12-05 19:49:19+01 vBook pdfm-bootstrap[6668]: inittool[6657]: * Guest /Users/Guest
2020-12-05 19:49:19+01 vBook pdfm-bootstrap[6671]: inittool[6657]: Current umask: 0022
2020-12-05 19:49:19+01 vBook pdfm-bootstrap[6682]: inittool[6657]: '/Applications/Parallels Desktop.app/Contents/MacOS/Parallels Service' - incorrect permissions (0755)
2020-12-05 19:49:19+01 vBook pdfm-bootstrap[6683]: inittool[6657]: Last bundle location file was not found
2020-12-05 19:49:19+01 vBook pdfm-bootstrap[6684]: inittool[6657]: Bundle '/Applications/Parallels Desktop.app' (16.1.1-49141) is not initialized: let's cook it
2020-12-05 19:49:19+01 vBook pdfm-bootstrap[6685]: inittool[6657]: Set bundle owners to 'root:wheel'
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6693]: inittool[6657]: Bootstrap actions
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6697]: inittool[6657]: Starting bootstrap action 'reg_bundles' in background
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6699]: inittool[6657]: Starting bootstrap action 'install_cli_links' in background
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6704]: inittool[6657]: Starting bootstrap action 'install_man_links' in background
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6708]: inittool[6657]: Starting bootstrap action 'install_sh_compl' in background
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6712]: inittool[6657]: Starting bootstrap action 'fix_library_parallels_dir_perms'
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6716]: inittool[6657]: Fixing Library/Parallels permissions...
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6722]: inittool[6657]: Bash comletion helpers will be installed into '/opt/local/etc/bash_completion.d'
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6747]: inittool[6657]: Endinging bootstrap action 'fix_library_parallels_dir_perms' with exit code '0'
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6749]: inittool[6657]: Starting bootstrap action 'setup_launchd_wrapper'
2020-12-05 19:49:20+01 vBook softwareupdated[6380]: Event handler called with flags: 103
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6759]: inittool[6657]: Endinging bootstrap action 'setup_launchd_wrapper' with exit code '0'
2020-12-05 19:49:20+01 vBook pdfm-bootstrap[6760]: inittool[6657]: Waiting background job 'reg_bundles' with pid 6698...
2020-12-05 19:49:21+01 vBook pdfm-bootstrap[6795]: inittool[6657]: 'reg_bundles' finished successfuly
2020-12-05 19:49:21+01 vBook pdfm-bootstrap[6796]: inittool[6657]: Waiting background job 'install_cli_links' with pid 6703...
2020-12-05 19:49:21+01 vBook pdfm-bootstrap[6797]: inittool[6657]: 'install_cli_links' finished successfuly
2020-12-05 19:49:21+01 vBook pdfm-bootstrap[6798]: inittool[6657]: Waiting background job 'install_man_links' with pid 6707...
2020-12-05 19:49:21+01 vBook pdfm-bootstrap[6799]: inittool[6657]: 'install_man_links' finished successfuly
2020-12-05 19:49:21+01 vBook pdfm-bootstrap[6800]: inittool[6657]: Waiting background job 'install_sh_compl' with pid 6711...
2020-12-05 19:49:21+01 vBook pdfm-bootstrap[6801]: inittool[6657]: 'install_sh_compl' finished successfuly
2020-12-05 19:49:21+01 vBook pdfm-bootstrap[6846]: inittool[6539]: Vendor info saved
Would installing version 15 of Parallels Desktop possibly succeed? But then, what about my existing VM, that already was launched with version 16? Would it work?