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As the machine learning and systems community strives to achieve higher energy-efficiency through custom DNN accelerators and model compression techniques, there is a need for a design space exploration framework that incorporates quantization-aware processing elements into the accelerator design space while having accurate and fast power, performance, and area models. In this work, we present QAPPA, a highly parameterized quantization-aware power, performance, and area modeling framework for DNN accelerators. Our framework can facilitate the future research on design space exploration of DNN accelerators for various design choices such as bit precision, processing element type, scratchpad sizes of processing elements, global buffer size, device bandwidth, number of total processing elements in the the design, and DNN workloads. Our results show that different bit precisions and processing element types lead to significant differences in terms of performance per area and energy. Specifically, our proposed lightweight processing elements achieve up to 4.9x more performance per area and energy improvement when compared to INT16 based implementation.