This paper proposes GeckoGripper, a novel soft, inflatable gripper based on the controllable adhesion mechanism of gecko-inspired micro-fiber adhesives, to pick-and-place complex and fragile non-planar or planar parts serially or in parallel. Unlike previous fibrillar structures that use peel angle to control the manipulation of parts, we developed an elastomer micro-fiber adhesive that is fabricated on a soft, flexible membrane, increasing the adaptability to non-planar three-dimensional (3D) geometries and controllability in adhesion. The adhesive switching ratio (the ratio between the maximum and minimum adhesive forces) of the developed gripper was measured to be around 204, which is superior to previous works based on peel angle-based release control methods. Adhesion control mechanism based on the stretch of the membrane and superior adaptability to non-planar 3D geometries enable the micro-fibers to pick-and-place various 3D parts as shown in demonstrations.