Optical computing has gained much attention due to its high speed, low energy consumption, and the fact that it is naturally parallelizable and multiplexable, etc. Single-bit optical logic gates based on a four-hidden-layer diffractive optical neural network (DONN) have been demonstrated with paired apertures. Here, we show a parallel-logic operation strategy based on two-hidden-layer DONN, showcasing their efficiency by multiple-bit (up to 16-bit) optical logic (e.g., NAND) operations. In addition, we demonstrate how NAND-DONN units can be utilized to achieve NOR and AND operations by flipping and cascading the DONN.