Multi-touch mice have been proposed to integrate the rich input vocabulary of multi-touch gestures into traditional computer mice. However, incorporating multi-finger gestures into a mouse introduces a design conflict: Mice are optimized for low friction to support smooth cursor navigation, whereas touch gestures require high friction for stability. In this paper, we present AutoPark—the concept of a multi-touch mouse that dynamically adjusts its friction based on user intent. AutoPark mouse operates in a low-friction mode for cursor movement and shifts to a high-friction state for multi-touch interactions. Our first user study measured the shear forces during touch gestures, confirming that conventional friction levels of mousepads are insufficient to maintain gesture input stability on a mouse. We developed an AutoPark prototype and conducted controlled experiments, comparing it with both conventional low-friction and gesture-favored high-friction setups. The results demonstrate that AutoPark enhances gesture stability while maintaining comparable pointing performance in our study tasks, supporting a richer gesture input vocabulary within mouse-based GUI interactions.