This paper introduces multivariate input-output models to predict the errors and bases dimensions of local parametric Proper Orthogonal Decomposition reduced-order models. We refer to these multivariate mappings as the MP-LROM models. We employ Gaussian Processes and Artificial Neural Networks to construct approximations of these multivariate mappings. Numerical results with a viscous Burgers model illustrate the performance and potential of the machine learning based regression MP-LROM models to approximate the characteristics of parametric local reduced-order models. The predicted reduced-order models errors are compared against the multi-fidelity correction and reduced order model error surrogates methods predictions, whereas the predicted reduced-order dimensions are tested against the standard method based on the spectrum of snapshots matrix. Since the MP-LROM models incorporate more features and elements to construct the probabilistic mappings they achieve more accurate results. However, for high-dimensional parametric spaces, the MP-LROM models might suffer from the curse of dimensionality. Scalability challenges of MP-LROM models and the feasible ways of addressing them are also discussed in this study.