The most commonly used two-dimensional filter banks are separable filter banks, which are obtained by cascading two one-dimensional (ID) filter banks in the form of a tree. The supports of the analysis and synthesis filters in the separable systems are the union of four rectangles. The natural nonseparable generalization of of such supports are the supports that are the union of four parallelograms. In this paper, we study the class of filter banks in which the supports of the analysis and synthesis filters are four parallelograms. We will study various types of support configurations for the four-parallelogram filter banks. Conditions on the configurations will be derived such that good design of analysis and synthesis filters are possible.