In this paper, we consider parametric Bayesian inference for stochastic differential equations driven by a pure-jump stable Lévy process, which is observed at high frequency. In most cases of practical interest, the likelihood function is not available; hence, we use a quasi-likelihood and place an associated prior on the unknown parameters. It is shown under regularity conditions that there is a Bernstein–von Mises theorem associated to the posterior. We then develop a Markov chain Monte Carlo algorithm for Bayesian inference, and assisted with theoretical results, we show how to scale Metropolis–Hastings proposals when the frequency of the data grows, in order to prevent the acceptance ratio from going to zero in the large data limit. Our algorithm is presented on numerical examples that help verify our theoretical findings.