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 Z2-valued spectral flow of an associated path of chiral skewadjoints. Furthermore, the related notion of Z2-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.