Inspired by experimental data, we investigate which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by demonstrating that the Weil polynomial of a hyperelliptic Jacobian must have a particular form modulo 2. For fixed g ≥ 1, the proportion of isogeny classes of g-dimensional abelian varieties defined over F q which fail this condition is 1 − Q(2g + 2)/2 g as q → ∞ ranges over odd prime powers, where Q(n) denotes the number of partitions of n into odd parts.