Authorization is challenging in distributed systems that cannot rely on the identification of nodes. Proof-of-work offers an alternative gate-keeping mechanism, but its probabilistic nature is incompatible with conventional security definitions. Recent related work establishes concrete bounds for the failure probability of Bitcoin's sequential proof-of-work mechanism. We propose a family of state replication protocols using parallel proof-of-work. Our bottom-up design from an agreement sub-protocol allows us to give concrete bounds for the failure probability in adversarial synchronous networks. After the typical interval of 10 minutes, parallel proof-of-work offers two orders of magnitude more security than sequential proof-of-work. This means that state updates can be sufficiently secure to support commits after one block (i.e., after 10 minutes), removing the risk of double-spending in many applications. We offer guidance on the optimal choice of parameters for a wide range of network and attacker assumptions. Simulations show that the proposed construction is robust against violations of design assumptions.