The temporal response of single-photon detectors is usually obtained by measuring their impulse response to short-pulsed laser sources. In this work, we present an alternative approach using time-correlated photon pairs generated in spontaneous parametric down-conversion (SPDC). By measuring the cross-correlation between the detection times recorded with an unknown and a reference photodetector, the temporal response function of the unknown detector can be extracted. Changing the critical phase-matching conditions of the SPDC process provides a wavelength-tunable source of photon pairs. We demonstrate a continuous wavelength-tunability from 526 nm to 661 nm for one photon of the pair, and 1050 nm to 1760 nm for the other photon. The source allows, in principle, to access an even wider wavelength range by simply changing the pump laser of the SPDC-based source. As an initial demonstration, we characterize single-photon avalance detectors sensitive to the two distinct wavelength bands, one based on Silicon, the other based on Indium Gallium Arsenide.