Multiple-output regression models require estimating multiple functions, one for each output. To improve parameter estimation in such models, methods based on structural regularization of the model parameters are usually needed. In this paper, we present a multiple-output regression model that leverages the covariance structure of the functions (i.e., how the multiple functions are related with each other) as well as the conditional covariance structure of the outputs. This is in contrast with existing methods that usually take into account only one of these structures. More importantly, unlike most of the other existing methods, none of these structures need be known a priori in our model, and are learned from the data. Several previously proposed structural regularization based multiple-output regression models turn out to be special cases of our model. Moreover, in addition to being a rich model for multiple-output regression, our model can also be used in estimating the graphical model structure of a set of variables (multivariate outputs) conditioned on another set of variables (inputs). Experimental results on both synthetic and real datasets demonstrate the effectiveness of our method.