Many modern recommender systems rely on matrix factorization techniques to produce personalized recommendations on the basis of the feedback that users provided on different items in the past. The feedback may take different forms, such as the rating of a movie, or the number of times a user listened to the songs of a given music band. Nonetheless, in some situations, the user can perform several actions on each item, and the feedback is multidimensional (e.g., the user of an e-commerce website can either click on a product, add the product to her cart or buy it). In this case, one can no longer view the recommendation problem as a matrix completion, unless the problem is reduced to a series of multiple independent problems, thus loosing the correlation between the different actions. In this case, the most suitable approach is to use a tensor approach to learn all dimensions of the feedback simultaneously. In this paper, we propose a specific instance of tensor completion and we show how it can be heavily par-allelized over both the dimensions (i.e., items, users, actions) and within each dimension (i.e., each item separately). We validate the proposed method both in terms of prediction accuracy and scalability to large datasets.