Multivariate areal data are common in many disciplines. When fitting spatial regressions for such data, one needs to account for dependence (both among and within areal units) to ensure reliable inference for the regression coefficients. Traditional multivariate conditional autoregressive (MCAR) models offer a popular and flexible approach to modeling such data, but the MCAR models suffer from two major shortcomings: (1) bias and variance inflation due to spatial confounding, and (2) high-dimensional spatial random effects that make fully Bayesian inference for such models computationally challenging. We propose the multivariate sparse areal mixed model (MSAMM) as an alternative to the MCAR models. Since the MSAMM extends the univariate SAMM, the MSAMM alleviates spatial confounding and speeds computation by greatly reducing the dimension of the spatial random effects. We specialize the MSAMM to handle zero-inflated count data, and apply our zero-inflated model to simulated data and to a large Census dataset for the state of Iowa.