Biological processes measured repeatedly among a series of individuals are standardly analyzed by mixed models. These biological processes can be adequately modeled by parametric Stochastic Differential Equations (SDEs). We focus on the parametric maximum likelihood estimation of this mixed-effects model defined by SDE. As the likelihood is not explicit, we propose a stochastic version of the Expectation-Maximization algorithm combined with the Particle Markov Chain Monte Carlo method. When the transition density of the SDE is explicit, we prove the convergence of the SAEM-PMCMC algorithm towards the maximum likelihood estimator. Two simulated examples are considered: an Ornstein-Uhlenbeck process with two random parameters and a time-inhomogeneous SDE (Gompertz SDE) with a stochastic volatility error model and three random parameters. When the transition density is unknown, we prove the convergence of a different version of the algorithm based on the Euler approximation of the SDE towards the maximum likelihood estimator.