  This note is about the topology of the path space of linear Fredholm
operators on a real Hilbert space. Fitzpatrick and Pejsachowicz introduced the
parity of such a path, based on the Leray-Schauder degree of a path of
parametrices. Here an alternative analytic approach is presented which reduces
the parity to the ${\mathbb Z}_2$-valued spectral flow of an associated path of
chiral skew-adjoints. Furthermore the related notion of ${\mathbb Z}_2$-index
of a Fredholm pair of chiral complex structures is introduced and connected to
the parity of a suitable path. Several non-trivial examples are provided. One
of them concerns topological insulators, another an application to the
bifurcation of a non-linear partial differential equation.
