Yes, buses do run from park to park among the four theme parks. There are, though, a couple points to keep in mind. Any PARK bus to or from the Magic Kingdom departs from the TTC, requiring a monorail or ferry boat ride across Seven Seas Lagoon (Buses from Disney resorts to hte MK drop off directly at the park).
There is no bus direct from DTD to any of the theme parks, which is why you had to go through the Contemporary last time. You have to transfer at some resort destination. What I do is arrange this transfer via a convenient resort, such as any of the monorail resorts to/from the MK, or Boardwalk for Epcot or the Studios (with a boat connection). You could also use the DTD boat to reach Port Orleans or Old Key West and transfer there.
In your specific case, take a bus from AK to the TTC, then boat or monorail to the MK.