In this paper, robust mean and covariance matrix estimation are considered in the context of mixed-effects models. Such models are widely used to analyze repeated measures data which arise in several signal processing applications that need to incorporate possible individual variations within a common behavior of individuals. In this context, most algorithms are based on the assumption that the observations follow a Gaussian distribution. Nevertheless, in certain situations in which the data set contains outliers, such assumption is not valid and leads to a dramatic performance loss. To overcome this drawback, we design an expectation-conditional maximization either algorithm in which the heterogeneous component is considered as a part of the complete data. Then, the proposed algorithm is cast into a parallel scheme w.r.t. the individuals in order to mitigate the computational cost and a possible central processor overload. Finally, the proposed algorithm is extended to deal with missing data which refers to the situation where part of the individual responses are unobserved. Numerical simulations are conducted to assess the performance of the proposed algorithm regarding robust regression estimators, probabilistic principal component analysis and its recent robust version.