In this paper, we present the results of calculations of cross-sections and rate coefficients for rotationally inelastic transitions in methanol (CH3OH), in its ground torsional state, induced by collisions with para-H-2 in its ground rotational state. The interaction potential was calculated by means of many-body perturbation theory, and the corresponding cross-sections computed using the coupled states approximation to the Schrodinger equation for the CH3OH-H-2 system. The rate coefficients have been calculated for kinetic temperatures in the range 5less than or equal toTless than or equal to200 K. We found that, although the rate coefficients for para-H-2 are qualitatively similar to those computed for He, the collisional propensity rules are less pronounced when the perturber is para-H-2 than when it is He. The rate coefficients for CH3OH-H-2 tend to be larger than for CH3OH-He, a tendency which is anticipated from a comparison of the corresponding interaction potentials. Although our discussion is in terms of E-type methanol, the analogous results for A-type methanol and para-H-2 are also available. Our calculations are the first to consider H-2 as a collision partner for CH3OH and are directly relevant to the interpretation of observations of methanol in the interstellar medium.