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?
Strictly speaking: no.
The posts table in the Wordpress database has the field post_parent, which only accommodates one other post id.
Tags may be a better solution if your site is set up to use them in that way -- what you're really doing here is categorizing.
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/
I have used to bSuite plug-in (http://maisonbisson/com/bsuite) to do what you describe. It has an "include" shortcode that will let you specify that the contents of one page are to be pulled from another page.
So, in Wordpress create two pages. Both named "Rental Assistance", one under "Programs" and another under "Housing". For the one under "Programs", but all your content. For the page under "Housing", have the only content of the page be
[include post_id="38" field="post_content"]