The Dimensio-HQ is the headquarters of the Dimensio-Agents. The Dimensio-HQ is where all of the Dimensio-Agents live in-between missions.
Please see the article Dimensio-Agents for a list of Dimensio-HQ's residents.
The HQ is told to be a large building. The rooms we've seen include a rooftop, several meeting rooms, a globe room, and a dark room in which the agents are debriefed by Commander FoX.
Parallel Kabookie and Parallel Zonis are in charge of a dungeon that contains various useless characters that were taken away from their original series. These characters are stored here to be used as a Dimensio-Army, and sacrificed in places where the Dimensio-Agents can't risk the lives of their real agents. See more on this location in the article DAHQ Dungeon.