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 sequence of integers related to the Bessel functions: equations 9 and 10. The sequence in this paper is the same sequence as in the OP, see OEIS:A002190.