I am developing a small site using Wordpress.org and would like multiple parents to share a child page. The purpose is that users coming to the site will for example associate "Rental Assistance" child page with the "Programs" parent and the "Housing" parent.
EDIT: I do not want to simply duplicate the content since this is redundant and poor design in the event of content changes. I have contemplated a redirect link, but that seems poor as well...
Is this possible?
No, use categories instead. Tags are similar, but categories are explicitly designed to be hierarchical. As anschauung said, you're categorizing, so use categories.
Details about the differences: http://support.wordpress.com/posts/categories-vs-tags/