The stabilization of a viscous interface stressed by an oscillating magnetic field is investigated. Account is taken of the influence of free-surface currents on the effective solidly rotating fluid column. Only azimuthal modes are considered in the linear perturbation. The dispersion relation with or without free-surface currents is obtained in the form of a linear Mathieu equation with complex coefficients. It is found that there is a nonlinear relation between the surface current density and both the stratified viscosity and the stratified azimuthal magnetic field. The surface currents disappear on the interface of the fluid column when the stratified magnetic field has the value of unity. At this value, a coupled parametric resonance occurs in the absence of angular velocity. The magnetic field plays a stabilizing role. This role increases with increasing surface currents. The angular velocity plays a destabilizing role, while the field frequency plays a stabilizing role and acts against the angular velocity. The stratified viscosity plays a damping role in the presence of the surface current density, while, in the absence of a surface current, it plays two opposite roles corresponding to the presence or absence of the field frequency. A set of graphs are used to illustrate the relation between the presence of free-surface currents and both the viscosity and the azimuthal magnetic field.