Parity conjecture for abelian surfaces

Assuming finiteness of the Tate–Shafarevich group, we prove that the Birch–Swinnerton-Dyer conjecture correctly predicts the parity of the rank of semistable principally polarised abelian surfaces. If the surface in question is the Jacobian of a curve, we require that the curve has good ordinary reduction at 2-adic places.