When I installed Ubuntu 16.04 under Parallels 12 by selecting New > Free Systems > Ubuntu, the name for the root user is automatically created as "parallels" during the installation. Is there a way to change this name for the root user during installation?
No. root user has name "root", and with random unknown password - as usual for Ubuntu distribution. "parallels" is another user with administrative privileges: it can gain root rights via special means like sudo(8). So you may login as "parallels" and create new administrative user with the name you like.