The parietal cortex (PC) is implicated in a confusing myriad of different cognitive processes/tasks. Consequently, understanding the nature and organization of the core underlying neurocomputations is challenging. According to the Parietal Unified Connectivity-biased Computation model, two properties underpin PC function and organization. Firstly, PC is a multidomain, context-dependent buffer of time- and space-varying input, the function of which, over time, becomes sensitive to the statistical temporal/spatial structure of events. Secondly, over and above this core buffering computation, differences in long-range connectivity will generate graded variations in task engagement across subregions. The current study tested these hypotheses using a group independent component analysis technique with two independent functional magnetic resonance imaging datasets (task and resting state data). Three functional organizational principles were revealed: Factor 1, inferior PC was sensitive to the statistical structure of sequences for all stimulus types (pictures, sentences, numbers); Factor 2, a dorsal-ventral variation in generally task-positive versus task-negative (variable) engagement; and Factor 3, an anterior-posterior dimension in inferior PC reflecting different engagement in verbal versus visual tasks, respectively. Together, the data suggest that the core neurocomputation implemented by PC is common across domains, with graded task engagement across regions reflecting variations in the connectivity of task-specific networks that interact with PC.