This paper proposes a new methodology to quantify the uncertainties associated to multiple dependent functional random variables, linked to a quantity of interest, called the covariate. The proposed methodology is composed of two main steps. First, the functional random variables are decomposed on a functional basis. The decomposition basis is computed by the proposed Simultaneous Partial Least Squares algorithm which enables to decompose simultaneously all the functional variables. Second, the joint probability density function of the coefficients of the decomposition associated to the functional variables is modelled by a Gaussian mixture model. A new method to estimate the parameters of the Gaussian mixture model based on a Lasso penalization algorithm is proposed. This algorithm enables to estimate sparse covariance matrices, in order to reduce the number of model parameters to be estimated. Several criteria are proposed to assess the efficiency of the methodology. Finally, its performance is shown on an analytical example and on a nuclear reliability test case.