I'm trying to create parent/child Single sections without using Structures. So in other words, create a Single section and assign it a parent Single section. Is this possible?
Take a look into Entry Types, I'm pretty sure that's a much better way to handle what you're trying to achieve:
So you'd have a Structure section (let's call it "Pages"), and within that Structure you have multiple Entry Types. You could have a distinct Entry Type for each page type ("About", "Contact", etc).
Once you've got that, it's easy to serve up different templates for each Entry Type:
Ignoring the obvious "use a Structure" answer, I suppose you could fake a "hierarchy" by placing a
parent entry field on each of your Single sections and manually assigning entries as relationships.
Then your Twig template could would just be checking those fields and building the family tree based on the relationships.