  For an elliptic curve with complex multiplication over a number field, the
$p^{\infty}$--Selmer rank is even for all $p$. \v{C}esnavi\v{c}ius proved this
using the fact that $E$ admits a $p$-isogeny whenever $p$ splits in the complex
multiplication field, and invoking known cases of the $p$-parity conjecture. We
give a direct proof, and generalise the result to abelian varieties.
