  A highlighting feature of Majorana bound states in two-dimensional
topological superconductors is that they gain a phase factor of $\pi$ upon
being orbited by a vortex. This work focuses on the vortex degree of freedom
itself and demonstrates that the change in the Majorana state is accompanied by
a fermion parity change within the vortex. Such a parity flip is interpreted as
a higher dimensional analog of the fermion parity pump mechanism in
superconducting wires as well as through general topological arguments. It is
demonstrated in terms of level crossings in three different situations - in i)
spin-triplet paired superconductors, and in proximity-induced superconducting
systems involving ii) quantum spin Hall-ferromagnet hybrids and iii) Chern
insulators.
