The Birch and Swinnerton–Dyer conjecture famously predicts that the rank of an elliptic curve can be computed from its L -function. In this article, we consider a weaker version of this conjecture called the parity conjecture and prove the following. Let E 1 and E 2 be two elliptic curves defined over a number field K whose 2-torsion groups are isomorphic as Galois modules. Assuming finiteness of the Shafarevich–Tate groups of E 1 and E 2 , we show that the Birch and Swinnerton-Dyer conjecture correctly predicts the parity of the rank of E 1 × E 2 . Using this result, we complete the proof of the p -parity conjecture for elliptic curves over totally real fields.