Processing moving object trajectories arises in many application domains and has been addressed by practitioners in the spatiotemporal database and Geographical Information System communities. In this work, we focus on a trajectory similarity search, the distance threshold query, which finds all trajectories within a given distance d of a search trajectory over a time interval. We demonstrate the performance of a multithreaded implementation which features the use of an R-tree index and which has high parallel efficiency (78%-90%). We introduce a GPGPU implementation which avoids the use of index-trees, and instead features a GPU-friendly indexing method. We compare the performance of the multithreaded and GPU implementations, and show that a speedup can be obtained using the latter. We propose two classes of algorithms, SetSplit and GreedySetSplit, to create efficient query batches that reduce memory pressure and computational cost on the GPU. However, we find that using fixed-size batches is sufficiently efficient in practice. We develop an empirical performance model for our GPGPU implementation that can be used to predict the response time of the distance threshold query. This model can be used to pick a good query batch size.