Parameter estimation in multinomial choice models becomes infeasible when the number of alternatives is large. As the parameters are alternative-specific, the number of parameters grows with the number of alternatives. Furthermore, in many applications the explanatory variables describe clusters which enter the model as large sets of dummies. This paper develops techniques for data-driven clustering over outcome categories and explanatory dummy parameters in a multinomial probit setting. A Dirichlet process mixture encourages parameters to cluster over the choice categories or explanatory categories, which favours a more parsimonious model without imposing any model restrictions. Simulation studies show that parameter clustering can improve greatly upon standard choice models in terms of predicting categories and fitting the category probabilities for many choice alternatives. The methods are applied to a high-dimensional choice data set of holiday destinations of a Dutch household panel.