Update 25 April 2020:
Many thanks to @KZL1992 @tomslominski @ptrofi @n-thumann for recent comments on their experiences with 20.04 and Parallels 13/14/15, especially the subsitution of the latest prl-tools-lin.iso for older Parallels to get it going.
First off, credit goes to github.com/rudolfratusinski for leading the way here.
https://gist.github.com/rudolfratusinski/a4d9e3caff11a4d9d81d2e84abc9afbf
In a very similar approach, copy the files from the Parallels installation media and drop them in a folder somewhere (eg. ~/parallels_fixed)
Go to the kmods directory (cd ~/parallels_fixed/kmods) and extract the files (tar -xzf prl_mod.tar.gz)
Remove prl_mod.tar.gz file from that directory (rm prl_mod.tar.gz)
Find this file: ~/<your-folder-goes-here>/kmods/prl_fs/SharedFolders/Guest/Linux/prl_fs/prlfs.h
Modify the file by going to line 16 and inserting a new line. Add this text: #include <uapi/linux/mount.h>
The file should now look like this. Save and exit.
..
#include <linux/fs.h>
#include <uapi/linux/mount.h>
#include <linux/types.h>
..
Go to the kmods directory (cd ~/parallels_fixed/kmods) and re-zip the files: tar -zcvf prl_mod.tar.gz . dkms.conf Makefile.kmods In case you missed it, yes that is a period(.) sitting there by itself and necessary.
Go to the installer directory cd ~/parallels_fixed/installer
Sudo chmod the script files: install-cli.sh (and others) to be executable eg. sudo chmod 777 *.sh
( @7feilee reports that if drag and drop is not working, then also try sudo chmod 777 prl* as the next step after chmod'ing the above script files ).
Then run that file with: sudo ./install-cli.sh -i --verbose
Reboot when it's finished.
@andyrudoff and @katter - many thanks for highlighting the file name correction.
@VirtualL - many thanks for highlighting it works on Ubuntu Mate 19.10 as well.
@Koongcen - many thanks for highlighting it works on Ubuntu 18.04 as well.
@BarryDeng - many thanks for highlighting it works on Ubuntu 20.04 as well.
Working well on 5.3.0-kali2-amd64 !