  We derive a local formula for the parity of the $2^{\infty}$-Selmer rank of
Jacobians of curves of genus $2$ or $3$ with a $K$-rational $2$-torsion point.
We give an explicit example to show how this local formula gives rank parity
predictions against which the $2$-parity conjecture may be tested. Our results
yield applications to the parity conjecture for semistable curves of genus $3$.
