Biological processes measured repeatedly among a series of individuals are standardly analyzed by mixed models. Recently, stochastic processes have been introduced to model the variability along time for each subject. Although the likelihood of these stochastic mixed models is intractable, various estimation methods have been proposed when the latent stochastic process is a discrete time finite state Markov chain. This is not the case when the hidden stochastic process is a continuous time process with non finite state space. This paper focuses on mixed models defined by parametric Stochastic Differential Equations (SDEs). We propose to use Particle MCMC algorithm for the maximum likelihood estimation of mixed SDE models, by combining it with SAEM algorithm. Theoretical and numerical convergence properties are discussed. Two simulated examples, an Ornstein-Uhlenbeck process and a time-inhomogeneous SDE with stochastic volatility, illustrate this estimator convergence, including the volatility parameter which is known to be hard to estimate.