Solution:

Half the integers from $0$ to $2007$ are odd, so each of $ad$ and $bc$ is odd with probability $\tfrac12 \cdot \tfrac12 = \tfrac14$ and even with probability $\tfrac34.$

The difference $ad - bc$ is even when both products have the same parity: $$\tfrac14 \cdot \tfrac14 + \tfrac34 \cdot \tfrac34 = \tfrac{1}{16} + \tfrac{9}{16} = \tfrac58.$$

Thus, the correct answer is E.