  The Hong-Ou-Mandel effect provides a mechanism to determine the
distinguishability of a photon pair by measuring the bunching rates of two
photons interfering at a beam splitter. Of particular interest is the
distinguishability in time, which can be used to probe a time delay. Photon
detectors themselves give some timing information, however--while that
resolution may dwarf that of an interferometric technique--typical analyses
reduce the interference to a binary event, neglecting temporal information in
the detector. By modelling detectors with number and temporal resolution we
demonstrate a greater precision than coincidence rates or temporal data alone
afford. Moreover, the additional information can allow simultaneous estimation
of a time delay alongside calibration parameters, opening up the possibility of
calibration-free protocols and approaching the precision of the quantum
Cram\'er-Rao bound.
