Okay, so, I moved my two VMs (Win 7 and Win 8) to a new MacBook Air. The Win 8 VM placed its apps on the dock on the new machine, but the Win 7 VM did not. Is there a way to force Parallels to add the apps back to the docK?
Do you mean the whole application folder or some particular apps? If application folder - then you need to open Virtual Machine configuration --> Options --> Applications and enable Applications sharing
If some particular applications icons - you will have to do it manually - start an application, right click an icon in the dock and choose @Keep in Dock@