The conjecture follows from the fact that, for any prime $p$ and any integers $m,r$, it holds that [1]
$$b_{mp^r}=b_m\mod(p)$$
$$b_m=0\mod(p)\;\;\text{if}\;\;p<m<2p.$$
Take $p=2$ and use $b_1=1$ to obtain the result that $b_m=1\mod 2$ if and only if $m$ is a power of 2.


[1] <A HREF="https://doi.org/10.1090/S0002-9939-1963-0166147-X">A sequence of integers related to the Bessel functions</A>: equations 9 and 10. The sequence in this paper is the same sequence as in the OP, see <A HREF="http://oeis.org/A002190">OEIS:A002190</A>.