We characterized the temporal dynamics of population activity in parietal cortex of monkeys as they solved a spatial cognitive problem posed by an object construction task. We applied pattern classification techniques to characterize patterns of activity coding object-centered side, a task-defined variable specifying whether an object component was located on the left or right side of a reference object, regardless of its retinocentric position. During a period in which the value of object-centered side, as defined by task events, remained constant, parietal cortex represented this variable using a dynamic neural code by activating neurons with the same spatial preference in rapid succession so that the pattern of active neurons changed dramatically while the spatial information they collectively encoded remained stable. Furthermore, if the neurons shared the same spatial preference, then their pretrial activity (measured before objects were shown) was correlated to a degree that scaled as a positive linear function of how close together in time the neurons would be activated later in the trial. Finally, we found that while parietal cortex represented task-critical spatial information using a dynamic neural code, it simultaneously represented task-irrelevant spatial information using a stationary neural code. These data demonstrate that dynamic spatial representations exist in parietal cortex, provide novel insight into the synaptic mechanisms that generate them, and suggest they may preferentially encode task-critical spatial information.