Single-photon detectors are essential for implementing optical quantum technologies, such as quantum key distribution, and for enhancing optical imaging systems such as lidar, while also playing a crucial role in studying the statistical properties of light. In this work, we show how the underlying photon statistics can be revealed by using a threshold detector, implemented as a Josephson parametric amplifier operating near a first-order phase transition. We describe the detection protocol, which utilizes a series of pumping pulses followed by the observation of activated switching events.