This paper analyzes the geometric quantities that remain unchanged during parallel mapping (i.e., mapping from a reference curved surface to a parallel surface with identical normal direction). The second gradient operator, the second class of integral theorems, the Gauss-curvature-based integral theorems, and the core property of parallel mapping are used to derive a series of parallel mapping invadants or geometri-cally conserved quantities. These include not only local mapping invadants but also global mapping invari-ants found to exist both in a curved surface and along curves on the curved surface. The parallel mapping invadants are used to identify important transformations between the reference surface and parallel surfaces. These mapping invadants and transformations have potential applications in geometry, physics, biome-chanics, and mechanics in which various dynamic processes occur along or between parallel surfaces.