The parent partition, also known as the root partition, is the partition from which you manage Hyper-V. Besides running child partitions or Virtual Machines, you can install and run applications in the parent partition as well. This however is a bad idea as the parent partition is unlike other Virtual Machines. It is a special Virtual Machine in that it is able to manage other Virtual Machines and it handles all IO of the other Virtual Machines. So technically it is possible to use it as a typical Virtual Machine but it is highly recommended to only use it for Hyper-V management.
The story changes when you reach the maximum allowed number of instances for the Windows Server 2008 edition you purchased. The rights you have with using the Windows Server 2008 edition is outlined in the Product Use Rights document. Product use Rights can be downloaded here in various languages.
When you reach the maximum allowed number of instances of the edition you purchased, you are not allowed to run arbitrary applications in the parent partition. The document states the following:
“If running the maximum allowed instances, the instance in the physical operating system environment may only be used to run hardware virtualization software, provide hardware virtualization services, or run software to manage and service operating system environments on the licensed server.”
In the case of Windows Server 2008 Enterprise Edition, you obtain the right to run one instance on the physical machine and four instances in Virtual Machines. When you reach the maximum of four Virtual Machines, the parent can basically only be used to manage those four Virtual Machines. Software in the parent partition for any other purpose than stated above may not be used.