Regular expression matching is essential for many applications, such as finding patterns in text, exploring substrings in large DNA sequences, or lexical analysis. However, sequential regular expression matching may be time-prohibitive for large problem sizes. In this paper, we describe a novel algorithm for parallel regular expression matching via deterministic finite automata. Furthermore, we present our tool PaREM that accepts regular expressions and finite automata as input and automatically generates the corresponding code for our algorithm that is amenable for parallel execution on shared-memory systems. We evaluate our parallel algorithm empirically by comparing it with a commonly used algorithm for sequential regular expression matching. Experiments on a dual-socket shared-memory system with 24 physical cores show speed-ups of up to 21x for 48 threads.