I've never worked on the Auto Park brake and don't know the details. It is hydraulically operated, so checking and periodically replacing the fluid (Dexron ATF) is one thing. Lubricating the moving mechanical parts is another. I understand there is also a cable that can freeze up or the end can break off.
Basically it is a drum type brake on the driveshaft that is applied by a spring that is always trying to set the brake. A hydraulic system pushes the spring back so you can move. If the hydraulic pump fails to run or build sufficient pressure, you are going to stay in park. Or if the drum brake itself seizes up or resists movement, it will stay either on or off, whichever it was at the time (usually on).
Here is one article on this beast (scan down to 2nd article on page)
2006 Tech and Projects
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL