AMC 10 2020a Problem 18

Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set ${0,1,2,3}.$ For how many such quadruples is it true that $a\cdot d-b\cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0\cdot 1-3\cdot 1 = -3$ is odd.) $\textbf{(A) } 48 \textbf{(B) } 64 \textbf{(C) } 96 \textbf{(D) } 128 \textbf{(E) } 192$