Parallels Desktop is software for your Mac to run Windows. Parallels Tools is software for your Windows that makes it run better with Parallels Desktop and Mac OS X. The Parallels Tools features are listed in the Parallels Tools Overview section of the help documentation.
http://download.parallels.com/desktop/v5/docs/en/Parallels_Desktop_Users_Guide/22272.htm
I'm not sure why the .html documentation isn't available on the web site for Parallels 6 but you can find it in the Help menu in Parallels Desktop. The web site includes the .pdf documentation and the .html documentation in .zip format but you can't google inside a zip file...:
http://www.parallels.com/products/desktop/technical-documentation/
If Parallels Tools is installed then
1) The virtual Machine menu shows "Reinstall Parallels Tools" instead of "Install Parallels Tools"
2) The tooltip for the Parallels Tools icon at the bottom right of the virtual machine window says "Parallels Tools are installed".
3) The bottom left of the virtual machine window won't say "Parallels Tools are not installed. From the Virtual Machine menu, choose install Parallels Tools."
4) "Add or Remove Programs" ("Programs and Features" in Windows 7) includes Parallels Tools.
5) In Windows Task Manager, you'll see Processes: coherence.exe (Parallels Coherence service), SharedIntApp.exe (Parallelse Shared Internet Applications Tool), prl_cc.exe (Parallels Control Center), prl_tools.exe (Parallels Tools), prl_tools_service.exe (Parallels Tools Service) (prl = Parallels just like the stuff you see in the Mac Activity Monitor)
6) In Computer Management -> Services (or Task Manager -> Services in Windows 7), you'll see Parallels Coherence Service (points to coherence.exe), and Parallels Tools Service (points to prl_tools_service.exe).
All the Parallels Tools are installed to C:\Program Files (x86)\Parallels\Parallels Tools. It includes lots of .dlls and drivers. The tools come from an .iso image installed on your Mac at /Library/Parallels/Tools/prl-tools-win.iso. I'm not sure which files are responsible for copying files between a Mac and VM window.
I don't think there's a way to know if it's installed correctly or not. You can try looking at the Parallels.log in Console.app to see what's happening when you drag a file to or from the VM.
What was the URL for that Parallels Troubleshooter Assistant topic?
If Parallels Shared Folders is working then you should see a list of folders when you double click the Parallels Shared Folder shortcut on the Windows desktop. The shortcut goes to \\psf\. Inside \\psf\ you should see Host if you have Shared Folders set to all disks, and Home. There may be other folders if you added specific individual shared folders. Host goes to / on your Mac (your Mac OS X disk). All your disks which are listed at /Volumes/ will be available in Windows at \\psf\Host\Volumes\. Home goes to ~/ on your Mac (your user folder in /Users/).