Maintaining a $k$-core decomposition quickly in a dynamic graph is an important problem in many applications, including social network analytics, graph visualization, centrality measure computations, and community detection algorithms. The main challenge for designing efficient $k$-core decomposition algorithms is that a single change to the graph can cause the decomposition to change significantly. We present the first parallel batch-dynamic algorithm for maintaining an approximate $k$-core decomposition that is efficient in both theory and practice. Given an initial graph with $m$ edges, and a batch of $B$ updates, our algorithm maintains a $(2 + \delta)$-approximation of the coreness values for all vertices (for any constant $\delta > 0$) in $O(B\log^2 m)$ amortized work and $O(\log^2 m \log\log m)$ depth (parallel time) with high probability. Our algorithm also maintains a low out-degree orientation of the graph in the same bounds. We implemented and experimentally evaluated our algorithm on a 30-core machine with two-way hyper-threading on $11$ graphs of varying densities and sizes. Compared to the state-of-the-art algorithms, our algorithm achieves up to a 114.52x speedup against the best multicore implementation and up to a 497.63x speedup against the best sequential algorithm, obtaining results for graphs that are orders-of-magnitude larger than those used in previous studies. In addition, we present the first approximate static $k$-core algorithm with linear work and polylogarithmic depth. We show that on a 30-core machine with two-way hyper-threading, our implementation achieves up to a 3.9x speedup in the static case over the previous state-of-the-art parallel algorithm.