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.
If you get a "Package manager is locked" error, you have to unlock apt.
This seems to allow me to install the default parallels tools package normally afterwards as well, no modding required.
At least in focal fossa, can't say the same for other distros as I haven't investigated it.