We demonstrate a new approach to measuring high-order temporal coherences that uses a four-element superconducting nanowire single-photon detector. The four independent, interleaved single-photon-sensitive elements parse a single spatial mode of an optical beam over dimensions smaller than the minimum diffraction-limited spot size. Integrating this device with four-channel time-tagging electronics to generate multi-start, multi-stop histograms enables measurement of temporal coherences up to fourth order for a continuous range of all associated time delays. We observe high-order photon bunching from a chaotic, pseudo-thermal light source, measuring maximum third- and fourth-order coherence values of 5.87 +/- 0.17 and 23.1 +/- 1.8, respectively, in agreement with the theoretically predicted values of 3! = 6 and 4! = 24. Laser light, by contrast, is confirmed to have coherence values of approximately 1 for second, third and fourth orders at all time delays.