Parent application is the application that loads another application. You'll notice that explorer.exe (the Windows shell or GUI if you will) is the primary one for most programs that run in Windows. Check them out by viewing the detailed descriptions of each Application Monitor rule. Other times you may have something like Yahoo Messenger as the parent application loading Internet Explorer (basically YM uses IE to connect to the internet).
By default, Application Monitor rules/alerts that are created are set to "learn" the parent (i.e. it auto-detects which program is loading which). You can also change or specify the parent application, but it's not necessary for average users who don't want to manually create rules.