Most research on count data regression models, i.e. models for there the dependent variable takes only non-negative integer values or count values, has focused on the univariate case. Very little attention has been given to joint modeling of two or more counts. We propose parametric regression models for bivariate counts based on squared polynomial expansions around a baseline density. The models are more flexible than the current leading bivariate count model, the bivariate Poisson. The models are applied to data on the use of prescribed and nonprescribed medications.