Chiplet-based architectures have been proposed to scale computing systems for deep neural networks (DNNs). Prior work has shown that for the chiplet-based DNN accelerators, the electrical network connecting the chiplets poses a major challenge to system performance, energy consumption, and scalability. Some emerging interconnect technologies such as silicon photonics can potentially overcome the challenges facing electrical interconnects as photonic interconnects provide high bandwidth density, superior energy efficiency, and ease of implementing broadcast and multicast operations that are prevalent in DNN inference. In this paper, we propose a chiplet-based architecture named SPRINT for DNN inference. SPRINT uses a global buffer to simplify the data transmission between storage and computation, and includes two novel designs: (1) a reconfigurable photonic network that can support diverse communications in DNN inference with minimal implementation cost, and (2) a customized dataflow that exploits the ease of broadcast and multicast feature of photonic interconnects to support highly parallel DNN computations. Simulation studies using ResNet50 DNN model show that SPRINT achieves 46% and 61% execution time and energy consumption reduction, respectively, as compared to other state-of-the-art chiplet-based architectures with electrical or photonic interconnects.