Sites are stopped in Internet Information Services (IIS) after a backup is completed with the Suspend domains until backup task is completed option enabled.
This is considered a software issue with ID PPPM-718 and has been fixed in Parallels Plesk 11.5.30 MU#36
Install Microupdate #36 to Parallels Plesk 11.5.30.
As a workaround (if you are running Plesk 11.0.9 and not able to upgrade for some reason), the Suspend domains until backup task is completed option can be disabled.
If the Suspend domains until backup task is completed option cannot be disabled, you can use Windows Task Scheduler to create a task that will start the domain in IIS and will be executed at the estimated time of backup completion:
Create an environment variable first. In "System Properties," go to "Environment Variables..." and add the new system variable:
Variable name: appcmd
Variable value: %windir%\\system32\\inetsrv\\AppCmd.exe
Then reboot the server. After that, you can create a task specifying the following settings:
Arguments: /c "%appcmd% start site example.com"