We propose a fully autonomous system for mobile robot exploration in unknown environments. Our system employs a novel frontier detection algorithm based on the fast front propagation (FFP) technique and uses parallel path planning to reach the detected front regions. Given an occupancy grid map in 2D, possibly updated online, our algorithm can find all the frontier points that can allow mobile robots to visit unexplored regions to maximize the exploratory coverage. Our FFP method is six~seven times faster than the state-of-the-art wavefront frontier detection algorithm in terms of finding frontier points without compromising the detection accuracy. The speedup can be further accelerated by simplifying the map without degrading the detection accuracy. To expedite locating the optimal frontier point, We also eliminate spurious points by the obstacle filter and the novel frontier region (FR) filter. In addition, we parallelize the global planning phase using the branch-and-bound A*, where the search space of each thread is confined by its best knowledge discovered during the parallel search. As a result, our parallel path-planning algorithm operating on 20 threads is about 32 times faster than the vanilla exploration system that operates on a single thread. Our method is validated through extensive experiments, including autonomous robot exploration in both synthetic and real-world scenarios. In the real-world experiment, we show that an autonomous navigation system using a human-sized mobile manipulator robot equipped with a low-end embedded processor that fully integrates our FFP and parallel path-planning algorithms.