To enable low-rank tensor completion and factorization, this paper puts forth a novel tensor rank regularization method based on the ℓ1,2-norm of the tensor's parallel factor analysis (PARAFAC) factors. Specifically, for an N-way tensor, upon collecting the magnitudes of its rank-1 components in a vector, the proposed regularizer controls the tensor's rank by inducing sparsity in the vector of magnitudes through ℓ1/N (pseudo)-norm regularization. Our approach favors sparser magnitude vectors than existing ℓ2/N- and ℓ1-based alternatives. With an eye towards large-scale tensor mining applications, we also develop efficient and highly scalable solvers for tensor factorization and completion using the proposed criterion. Extensive numerical tests using both synthetic and real data demonstrate that the proposed criterion is better in terms of revealing the correct number of components and estimating the underlying factors than competing alternatives.