Solution:

The first die is odd (a $1$ or $3$) with probability $\dfrac23$ and even with probability $\dfrac13.$ The second die is odd (a $5$) with probability $\dfrac13$ and even with probability $\dfrac23.$

The sum is odd when the two parities differ: $$\dfrac13 \cdot \dfrac13 + \dfrac23 \cdot \dfrac23 = \dfrac19 + \dfrac49 = \dfrac59.$$

Thus, D is the correct answer.