The long propagation delay of underwater acoustic communications has made traditional handshaking-based medium access control (MAC) protocols inefficient for underwater sensor networks (UWSNs). We envision that this unique feature, although often taken as negative, can be leveraged to improve the parallelism between multiple senders, i.e., transmitting data simultaneously so as to achieve higher channel utilization. In this paper, we propose a novel Delay-Aware Probability-based MAC protocol, DAP-MAC, which eliminates the handshaking process and utilizes concurrent transmissions to significantly improve the network throughput. Compatibility relation among senders based on their distances to the common receiver is derived at run-time. A utility-optimization framework uses this compatibility relation to determine the best transmission strategies for senders. Our extensive simulations demonstrate that DAP-MAC achieves better system throughput than the existing representative underwater MAC protocols.