It looks like zPanel's installer checks if you're running Ubuntu 12.04, but does not care if you're running e.g. 12.04.5. zPanel itself should work on Ubuntu 12.04.5 so I would suggest removing the "12.04" check in the installer.
Download the installer:
wget -O install_zpanel.sh https://raw.githubusercontent.com/zpanel/installers/master/install/Ubuntu-12_04/10_1_1.sh
chmod +x install_zpanel.sh
Then, open it in an editor and delete lines 61-66:
if [ "$OS" = "Ubuntu" ] && [ "$VER" = "12.04" ]; then
echo "Sorry, this installer only supports the installation of ZPanel on Ubuntu 12.04."
Or, run the following command to do that:
sed -ie '61,66d' install_zpanel.sh
You should now be able to install zPanel: