Wordpress: Can one page have two parents?
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/
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.
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"]