For a geometric global p-adic Galois representation $V$, the $p$-parity conjecture asserts the relation between the parity of the Selmer rank and the root number attached to $V$. In the case where $V$ is the $p$-adic Tate module of an elliptic curve, this may be viewed as a mod $2$ analogue of the Birch and Swinnerton-Dyer conjecture, conditional on the finiteness of the Tate–Shafarevich group. In this talk, I will survey the conjecture, as well as a local analogue for rank-two representations. This is joint work with A. Burungale, K. Nakamura, and K. Ota.