  We study the properties of a topological Josephson junction made of both
edges of a 2D topological insulator. We show that, due to fermion parity
pumping across the bulk, the global parity of the junction has a clear
signature in the periodicity and critical value of the Josephson current. In
particular, we find that the periodicity with the flux changes from $4\pi$ in a
junction with an even number of quasi-particles to $2\pi$ in the odd sector. In
the case of long junctions, we exhibit a rigorous mathematical connection
between the spectrum of Andreev bound-states and the fermion parity anomaly,
through bosonization. Additionally, we discuss the rather quantitative effects
of Coulomb interactions on the Josephson current.
