When strongly pumped at twice their resonant frequency, nonlinear resonators develop a high-amplitude intracavity field, a phenomenon known as parametric self-oscillations. The boundary over which this instability occurs can be extremely sharp and thereby presents an opportunity for realizing a detector. Here, we operate such a device based on a superconducting microwave resonator whose nonlinearity is engineered from kinetic inductance. The device indicates the absorption of low-power microwave wavepackets by transitioning to a self-oscillating state. Using calibrated pulses, we measure the detection efficiency to zeptojoule energy wavepackets. We then apply it to measurements of electron spin resonance, using an ensemble of 209Bi donors in silicon that are inductively coupled to the resonator. We achieve a latched readout of the spin signal with an amplitude that is five hundred times greater than the underlying spin echoes.