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Sparse matrix multiplication (SpGEMM) is a fundamental kernel used in many diverse application areas, both numerical and discrete. For example, many algebraic graph algorithms rely on SpGEMM in the tropical semiring to compute shortest paths in graphs. Recently, SpGEMM has received growing attention regarding implementations for specific (parallel) architectures. Yet, this concerns only the static problem, where both input matrices do not change. In many applications, however, matrices (or their corresponding graphs) change over time. Although recomputing from scratch is very expensive, we are not aware of any dynamic SpGEMM algorithms in the literature. In this paper, we thus propose a batch-dynamic algorithm for MPI-based parallel computing. Building on top of a distributed graph/matrix data structure that allows for fast updates, our dynamic SpGEMM reduces the communication volume significantly. It does so by exploiting that updates change far fewer matrix entries than there are non-zeros in the input operands. Our experiments with popular benchmark graphs show that our approach pays off. For batches of insertions or removals of matrix entries, our dynamic SpGEMM is substantially faster than the static algorithms in the state-of-the-art competitors CombBLAS, CTF and PETSc.