The origin of the parabolic background of magnetoresistance oscillations measured in finite-width superconducting mesoscopic rings with input and output stubs and in patterned films is analyzed. The transmission model explaining the sinusoidal oscillation of magnetoresistance is extended to address the parabolic background as a function of the magnetic field. Apart from the interference mechanism activated by the ring, pinned superconducting vortices as topological defects introduce a further interference-based distribution of supercurrents that affects, in turn, the voltmeter-sensed quasiparticles. The onset of vortices changes the topology of the superconducting state in a mesoscopic ring in such a way that the full magnetoresistance dynamics can be interpreted due to the interference of the constituents of the order parameter induced by both the ring with its doubly connected topology and the vortex lattice in it.