I used to run Ubuntu 14.04 on parallels and everything was fine. When I upgraded to 16.04 it got stuck on
/dev/sda1: clean XXXXXX/XXXXXX files, XXXXX/XXXXX blocks
and now it won't boot.
First, your files are intact, and your system is not "stuck." X-Windows is just not starting. As I said in a comment, you could use CTRL-ALT-F1 or SSH to log into the Parallels instance.
You need to update manually the Parallels tool, which fixes the video driver. You can follow the directions in this KB article. (It's for an older version of Ubuntu and Parallels, but the steps are the same.) http://kb.parallels.com/en/118776
In summary, you attach the "parallels tools linux iso" (it's in the Parallels.app bundle) to the virtual CD-ROM. Start the virtual machine. Use CTRL-ALT-F1 to get to a text terminal, login, mount the virtual cd, and run the install script. Once reboot, X-Windows will start correctly.
I had the same problem when upgrade from 14.04 but install
Xenial will resolve it, got it from apt: relocation error: version GLIBCXX_3.4.21 not defined in file libstdc++.so.6 with link time reference