Abstract QR factorization is a popular calculation method in matrix algebra due to its usefulness in the solution of problems such as estimating least squares and calculating eigenvalues. In this paper, we describe a parallel algorithm for the calculation of the QR factorization on a hypercube architecture of the SIMD type with distributed memory. We have chosen the modified Gram-Schmidt method with pivoting to determine the QR factorization as it is characterized by good numerical stability. As an application of the QR factorization, we analyze the problem of least squares, developing a complementary parallel algorithm for solving it. Both algorithms are general; they are not limited by the size of the problem or the dimension of the hypercube. Finally, we analyze the algorithmic complexities of both parallel algorithms.