In this paper we consider the parameter estimation problem associated to partially-observed time changed SDEs, with observations that are given at discrete times. In particular we consider both likelihood and Bayesian estimation. We develop new Markov chain Monte Carlo (MCMC) algorithms which allow an unbiased score-based stochastic approximation method to provide likelihood-type parameter estimators. We also use a variant of this MCMC algorithm to perform multilevel-based Bayesian parameter estimation. We prove that this latter method achieves a mean square error of $\mathcal{O}(ε^2)$ ($ε>0$) with a cost of $\mathcal{O}(ε^{-2}\log(ε)^2)$. Our methodologies are tested numerically on both simulated and real data.

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