In this paper, we propose a single-actuator gripper that can lift thin objects lying on a flat surface, in addition to the ability as a standard parallel gripper. The key is a crawler on the fingertip, which is underactuated together with other finger joints and switched with a passive and spring-loaded mechanism. While the idea of crawling finger is not a new one, this paper contributes to realize the crawling without additional motor. The gripper can passively change the mode from the parallel approach mode to the pull-in mode, then finally to the power grasp mode, according to the grasping state. To optimize the highly underactuated system, we take a combination of black-box optimization and physics simulation of the whole grasp process. We show that this simulation-based approach can effectively consider the precontact motion, in-hand manipulation, power grasp stability, and even failure mode, which is difficult for the static-equilibrium-analysis-based approaches. In the last part of the paper, we demonstrate that a prototype gripper with the proposed structure and design parameters optimized under the proposed process successfully power-grasped a thin sheet, a softcover book, and a cylinder lying on a flat surface.