2020 AMC 10A 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 \qquad \textbf{(B) } 64 \qquad \textbf{(C) } 96 \qquad \textbf{(D) } 128 \qquad \textbf{(E) } 192

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.