The registration is not really the issue when it comes to using apps. That only effects support and warrantee.
In order to get at your dad's app you need to login to the App Store on the iPad with your Dad's Apple ID. Once that is done he can see all his apps under the Purchases icon. Download the ones he wants. If you don't see a Purchased section in the store, then search for the app(s). Re-downloading them (as long as you are using the right account) will not cost anything.
Even easier would be to restore the iPad, and re-initialize it in iTunes using your dad's account. You will lose any apps that you loaded onto the iPad, but unless they are expensive or important for other reasons the expense may be worth it.
Otherwise every time an update for an app comes out you will have to change your App Store login to the account that purchased the app to install that update.
Normally keeping apps when transferring an iPad to a new owner is not legal. However Apple does allow it for immediate family in their EULA. Up to you to decide if you qualify, and/or if it is worth the inconvenience of having two switch accounts back and forth. Not to mention trusting your dad not to accidentally purchase stuff on your account.
