State-machine replication, a fundamental approach to designing fault-tolerant services, requires commands to be executed in the same order by all replicas. Moreover, command execution must be deterministic: each replica must produce the same output upon executing the same sequence of commands. These requirements usually result in single-threaded replicas, which hinders service performance. This paper introduces Parallel State-Machine Replication (P-SMR), a new approach to parallelism in state-machine replication. P-SMR scales better than previous proposals since no component plays a centralizing role in the execution of independent commands---those that can be executed concurrently, as defined by the service. The paper introduces P-SMR, describes a "commodified architecture" to implement it, and compares its performance to other proposals using a key-value store and a networked file system.