Data-driven models of functional magnetic resonance imaging (fMRI) activity can elucidate dependencies that involve the combination of multiple brain regions. Activity in some regions during resting-state fMRI can be predicted with high accuracy from the activities of other regions. However, it remains unclear in which regions activity depends on unique integration of multiple predictor regions. To address this question, sparse (parsimonious) models could serve to better determine key interregional dependencies by reducing false positives. We used resting-state fMRI data from 46 subjects, and for each region of interest (ROI) per subject we performed whole-brain recursive feature elimination (RFE) to select the minimal set of ROIs that best predicted activity in the modeled ROI. We quantified the dependence of activity on multiple predictor ROIs, by measuring the gain in prediction accuracy of models that incorporated multiple predictor ROIs compared to models that used a single predictor ROI. We identified regions that showed considerable evidence of multiregional integration and determined the key regions that contributed to their observed activity. Our models reveal fronto-parietal integration networks, little integration in primary sensory regions, as well as redundancy between some regions. Our study demonstrates the utility of whole-brain RFE to generate data-driven models with minimal sets of ROIs that predict activity with high accuracy. By determining the extent to which activity in each ROI depended on integration of signals from multiple ROIs, we find cortical integration networks during resting-state activity.