I'm afraid you are missing independent events with incompatible events.
There are several equivalent definitions of independent events. A quite useful one is that two events are independent if knowing that one of them has happened doesn't give any information about the probability of the other.
Here, if A happens, B has not happened - that is, they are incompatible. Then, knowing that A has happened gives a lot of information about the probability of B: if you know the dice yielded 1, 3 or 5, you can be sure that it didn't yield 2, 4 or 6. Therefore, they aren't independent.
Using other common definitions of independence lead to the same conclusion that A and D are not independent (definitions given assuming events $A$ and $D$ are non empty with probability different from 0):
$P(A \cap D) = P(A)·P(D)$
This definition is not fulfilled because $P(A \cap D) = 0$ but $P(A)·P(D)=0.5·0.5=0.25$
Another common definition is that for independent events $P(A|D)=P(A)$ and $P(D|A)=P(D)$, and here $P(A|D)=0$ while $P(A)=0.5$ (and the same for D).
Please notice that the same reasoning can be used with any pair of incompatible events: incompatible events with probability different than zero are never independent.