Although legged robots have demonstrated effective mobility in some natural settings, as robot size decreases, obstacles in their environment become challenging to overcome. Small arthropods scale obstacles many times their size through jumps powered by mechanisms that overcome speed and power limitations of muscle alone. The motivation for this study was to explore the marriage of impulsive (jumping) and nonimpulsive (cyclic legged ambulation) behaviors in a centimeter-scale robot. Here, jumping is achieved by striking the ground with a bioinspired appendage connected to a parallel linkage. As the linkage configuration passes through the singularity, a torque reversal occurs whereby elastic energy slowly stored by force-dense velocity-limited shape memory alloy actuators is rapidly released. A passively driven elastic hinge is introduced in the striking arm to mediate ground contact forces and direct jumping. High-speed video recording of the 14-millisecond launch phase reveals previously undocumented takeoff dynamics closely resembling those of springtails. A dynamic model was derived, and an experimentally validated simulation was used to optimize the design of key components. The 2.2-gram, 6.1-centimeter-long mechanism achieved a maximum horizontal jumping distance of 1.4 meters (23 body lengths), surpassing that of similarly sized insects. The mechanism was integrated with an agile quadrupedal microrobot with leg articulation suitable to achieve the ideal jumping posture. The platform demonstrated repeatable directional takeoffs and upright landings, enabling complex maneuvers to overcome obstacles and gaps. Last, we used this bioinspired robot to offer reflection on hypotheses related to springtail jumping behavior.