Q: If I upgrade to Mountain Lion, will my Windows based apps still work or will I have to reinstall them?

Windows-based apps don't run on Macs, using Mac OS X, unless you are using a "virtual machine" environment, such as VMware Fusion or Parallels Desktop.  Then, the Windows OS is running separately, as a virtual machine, so it should not be affected (directly) by a Mac OS X update.  Windows applications that ran before should still run.  The virtual machine environment (VMware Fusion, Parallels Desktop, or other) just needs to run properly under Mountain Lion.

OR, if you are using Apple's BootCamp to start up directly into Window, that should not be affected in any way from an update to Mac OS X.  As long as BootCamp is able to start up Windows, it is running Windows only (not Mac OS X), so an upgrade to Mountain Lion should have no affect.

As I said, the Windows applications are running in the virtual machine, using the Windows system.  It is not directed affected by upgrading Mac OS X to Mountain Lion.  So you should not need to re-install anything that runs in the Windows virtual machine.

The real question is, does Parallel Desktop, which is the Mac OS X application, work properly under Mountain Lion?  You should go to

http://www.parallels.com/products/desktop/

and make sure you are using the latest version that has been updated to work with Mountain Lion.

to add,

while the VMs are self-contained, and you need to the latest version of parallels (or upgrade to it) for Mountain Lion, Windows IN the VMs may report new hardware (it does when you move VMs to new hardware, but usually doesn't if the OS is upgraded. In your case, you may upgrade Parallels though).

Be aware that Windows hence may need to re-authorize if it detects being on 'new hardware'.