  Hirschhorn and Sellers studied arithmetic properties of the number of
partitions with odd parts distinct. In another direction, Hammond and Lewis
investigated arithmetic properties of the number of bipartitions. In this
paper, we consider the number of bipartitions with odd parts distinct. Let this
number be denoted by $pod_{-2}(n)$. We obtain two Ramanujan type identities for
$pod_{-2}(n)$, which imply that $pod_{-2}(2n+1)$ is even and $pod_{-2}(3n+2)$
is divisible by 3. Furthermore, we show that for any $\alpha\geq 1$ and $n\geq
0$, $ pod_{-2}(3^{2\alpha+1}n+\frac{23\times 3^{2\alpha}-7}{8})$ is a multiple
of 3 and $pod_{-2}(5^{\alpha+1}n+\frac{11\times 5^\alpha+1}{4})$ is divisible
by 5. We also find combinatorial interpretations for the two congruences modulo
2 and 3.
