In this paper, we consider the problem of Bayesian filtering and inference for time series data modeled as noisy, discrete-time observations of a stochastic differential equation (SDE) with undetermined parameters. We develop a Metropolis algorithm to sample from the high-dimensional joint posterior density of all SDE parameters and state time series. Our approach relies on an innovative density tracking by quadrature (DTQ) method to compute the likelihood of the SDE, the part of the posterior that requires the most computational effort to evaluate. As we show, the DTQ method lends itself to a natural implementation using Scala and Apache Spark, an open source framework for scalable data mining. We study the performance and scalability of our algorithm on filtering and inference problems for both regularly and irregularly spaced time series.