Assume you have a network as follow (where X->Y implies X is the parent of Y)
A->D, B->D
How does one compute $P(A,B|D)$?
A and B are independent so my intuition tells me
$P(A,B|D)= P(A|D)\times P(B|D) = [\frac{P(D|A)}{P(A)}]\times[\frac{P(D|B)}{P(B)}]$
but I am not sure.
No. You cannot infer conditional independence from pairwise independence.
Applying Bayes' Rule can only obtain:
$$\begin{align} \mathsf P(A, B\mid D) ~=~& \dfrac{\mathsf P(D\mid A, B)~\mathsf P(A,B)}{\mathsf P(D)} \\[1ex] =~& \dfrac{\mathsf P(D\mid A, B)~\mathsf P(A)~\mathsf P(B)}{\mathsf P(D)} \end{align}$$