How can I save the parent category/project of a saved task or project?

Go to the saved items overlay through the menu at the top right. 

Find the item and click the edit icon (pencil) on the right. 

Then edit the title and add the #category shortcut to it. 

So if you want to have a task called "Order groceries" always be filed in the "Household" category, you would modify the title to "Order groceries #Household".