Many modern recommender systems rely on matrix factor-ization techniques to produce personalized recommendationson the basis of the feedback that users provided on differ-ent items in the past. The feedback may take different forms,such as the rating of a movie, or the number of times a userlistened to the songs of a given music band. Nonetheless, insome situations, the user can perform several actions on eachitem, and the feedback is multidimensional (e.g., the user ofan e-commerce website can either click on a product, add theproduct to her cart or buy it). In this case, one can no longerview the recommendation problem as a matrix completion,unless the problem is reduced to a series of multiple inde-pendent problems, thus loosing the correlation between thedifferent actions. In this case, the most suitable approach is touse a tensor approach to learn all dimensions of the feedbacksimultaneously. In this paper, we propose a specific instanceof 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). Wevalidate the proposed method both in terms of prediction ac-curacy and scalability to large datasets.