  We study the parity of 2-Selmer ranks in the family of quadratic twists of a
fixed principally polarised abelian variety over a number field. Specifically,
we determine the proportion of twists having odd (resp. even) 2-Selmer rank.
This generalises work of Klagsbrun--Mazur--Rubin for elliptic curves and Yu for
Jacobians of hyperelliptic curves. Several differences in the statistics arise
due to the possibility that the Shafarevich--Tate group (if finite) may have
order twice a square. In particular, the statistics for parities of 2-Selmer
ranks and 2-infinity Selmer ranks need no longer agree and we describe both.
