Solution:

Let $p$ be the probability that an odd number is rolled. Then $3p$ is the probability an even number is rolled. We know that $$p + 3p = 1 \Rightarrow p = \dfrac{1}{4}.$$

The only way for the sum to be even is if both rolls have the same parity. This happens with a probability of $$\dfrac{1}{4}^2 + \dfrac{3}{4}^2 = \dfrac{10}{16} = \dfrac{5}{8}.$$

Thus, E is the correct answer.