When an airport use two runways that are parallel at the same time, my question is which tower controller owns the "ground space" between the runways?
Typically, neither. A separate controller, "___ Ground," controls taxiing traffic. On a big enough airport such as Denver (KDEN) or Atlanta (KATL), there may be more than one ground control, each on its own frequency. Their domains are split geographically: one controls east (DEN) or north (ATL) of the terminals, while the other controls west or south.
At some airports, though, a single Tower frequency owns both parallel runways, for example Austin TX (KAUS), and in San Francisco, one controller works all four runways (1L/R and 28L/R). No rookies working that position!