We present a set of parallel algorithms for computing exact k-nearest neighbors in low dimensions. Many k-nearest neighbor algorithms use either a kd-tree or the Morton ordering of the point set; our algorithms combine these approaches using a data structure we call the \textit{zd-tree}. We show that this combination is both theoretically efficient under common assumptions, and fast in practice. For point sets of size (n) with bounded expansion constant and bounded ratio, the zd-tree can be built in (O(n)) work with (O(n^{\epsilon})) span for constant (\epsilon<1), and searching for the (k)-nearest neighbors of a point takes expected (O(klog k)) time. We benchmark our k-nearest neighbor algorithms against existing parallel k-nearest neighbor algorithms, showing that our implementations are generally faster than the state of the art as well as achieving 75x speedup on 144 hyperthreads. Furthermore, the zd-tree supports parallel batch-dynamic insertions and deletions; to our knowledge, it is the first k-nearest neighbor data structure to support such updates. On point sets with bounded expansion constant and bounded ratio, a batch-dynamic update of size (k) requires (O(k log n/k)) work with (O(k^{\epsilon} + \text{polylog}(n))) span.