Given that $a$ and $b$ are integers.
And $a+a^{2}b^{3}$ is odd
So, $a(1+ab^{3})$ is also odd.
We now that only odd$\times$ odd is odd. (even $\times$ even = even, odd $\times$ even = even)
Therefore $a$ is odd and $(1+ab^{3})$ is odd
So, $ab^{3}=$ even $\{\because 1+x \text{ is odd}\implies x \text{ is } odd-1\implies x \text{ is even}\}$
Since $a$ is odd and $ab^{3}$ is even, $b$ must be even.
Therefore, $a$ is odd and $b$ is even.
Hence, the correct answer is (D).