There are numerous applications which involve modeling multi-dimensional count data. When such data exhibit an excess of zeros, common count models are no longer adequate. In this work, we propose a new bivariate zero-inflated Poisson model appropriate for modeling multivariate counts with a surplus of zeros. The proposed model construction is based on a mixture model approach involving a common mass at zero along with a Poisson random pair. The latter stems from a bivariate Poisson model wherein a positive correlation is induced via a comonotonic shock. The properties of the proposed model are detailed, and several estimation methods are explored. Comprehensive simulations as well as a real data illustration demonstrate the practical applications of the proposed model.