Maintaining spatial data (points in two or three dimensions) is crucial and has a wide range of applications, such as graphics, GIS, and robotics. To handle spatial data, many data structures, called spatial indexes, have been proposed, e.g. kd-trees, oct/quadtrees (also called Orth-trees), R-trees, and bounding volume hierarchies (BVHs). In real-world applications, spatial datasets tend to be highly dynamic, requiring batch updates of points with low latency. This calls for efficient parallel batch updates on spatial indexes. Unfortunately, there is very little work that achieves this. In this paper, we systematically study parallel spatial indexes, with a special focus on achieving high-performance update performance for highly dynamic workloads. We select two types of spatial indexes that are considered optimized for low-latency updates: Orth-tree and R-tree/BVH. We propose two data structures: the P-Orth tree, a parallel Orth-tree, and the SPaC-tree family, a parallel R-tree/BVH. Both the P-Orth tree and the SPaC-tree deliver superior performance in batch updates compared to existing parallel kd-trees and Orth-trees, while preserving better or competitive query performance relative to their corresponding Orth-tree and R-tree counterparts. We also present comprehensive experiments comparing the performance of various parallel spatial indexes and share our findings at the end of the paper.