  For an element $g$ in a group $X$, we say that $g$ has 2-part order $2^{a}$
if $2^{a}$ is the largest power of 2 dividing the order of $g$. We prove lower
bounds on the proportion of elements in finite classical groups in odd
characteristic that have certain 2-part orders. In particular, we show that the
proportion of odd order elements in the symplectic and orthogonal groups is at
least $C/\ell^{3/4}$, where $\ell$ is the Lie rank, and $C$ is an explicit
constant. We also prove positive constant lower bounds for the proportion of
elements of certain 2-part orders independent of the Lie rank. Furthermore, we
describe how these results can be used to analyze part of Yal\c{c}inkaya's
Black Box recognition algorithm for finite classical groups in odd
characteristic.
