In this paper, we investigate the parallel surfaces of all three types of surfaces of revolution in the Galilean 3-space $G^3$ under the condition that the mean curvature of the parallel surface is constant. For each type of revolution surface, we explicitly compute the mean curvature of the corresponding parallel surface and derive the ordinary differential equations governing the profile curves. By solving these equations, we obtain explicit expressions for the functions defining these surfaces. Furthermore, several representative examples are presented, and the resulting surfaces are visualized to illustrate the influence of the parallel surface transformation on the geometry of revolution surfaces in the Galilean setting.