Inspired by human vision, we propose a new periphery-fovea multi-resolution driving model that predicts vehicle speed from dash camera videos. The peripheral vision module of the model processes the full video frames in low resolution with large receptive fields. Its foveal vision module selects sub-regions and uses high-resolution input from those regions to improve its driving performance. We train the fovea selection module with supervision from driver gaze. We show that adding high-resolution input from predicted human driver gaze locations significantly improves the driving accuracy of the model. Our periphery-fovea multi-resolution model outperforms a uni-resolution periphery-only model that has the same amount of floating-point operations. More importantly, we demonstrate that our driving model achieves a significantly higher performance gain in pedestrian-involved critical situations than in other non-critical situations. Our code is publicly available at https://github.com/pascalxia/periphery-fovea-driving.