For an abelian variety A over a number field K, a consequence of the Birch and Swinnerton-Dyer conjecture is the 2-parity conjecture: the global root number agrees with the parity of the 2-infinity Selmer rank. It is a standard result that the root number may be expressed as a product of local terms and we show that, over any quadratic extension of K, the same holds true for the parity of the 2-infinity Selmer rank. Using this we prove several new instances of the 2-parity conjecture for general principally polarised abelian varieties by comparing the local contributions arising. Somewhat surprisingly, the local comparison relies heavily on results from the theory of quadratic forms in characteristic 2.