We fit 54,296 sparsely-sampled asteroid lightcurves in the Palomar Transient Factory (PTF) survey to a combined rotation plus phase-function model. Each lightcurve consists of 20 or more observations acquired in a single opposition. Using 805 asteroids in our sample that have reference periods in the literature, we find the reliability of our fitted periods is a complicated function of the period, amplitude, apparent magnitude and other lightcurve attributes. Using the 805-asteroid ground-truth sample, we train an automated classifier to estimate (along with manual inspection) the validity of the remaining ∼53,000 fitted periods. By this method we find 9,033 of our lightcurves (of ∼8,300 unique asteroids) have 'reliable' periods. Subsequent consideration of asteroids with multiple lightcurve fits indicate a 4% contamination in these 'reliable' periods. For 3,902 lightcurves with sufficient phase-angle coverage and either a reliably-fit period or low amplitude, we examine the distribution of several phase-function parameters, none of which are bimodal though all correlate with the bond albedo and with visible-band colors. Comparing the theoretical maximal spin rate of a fluid body with our amplitude versus spin-rate distribution suggests that, if held together only by self-gravity, most asteroids are in general less dense than ∼2 g/cm 3 , while C types have a lower limit of between 1 and 2 g/cm 3 . These results are in agreement with previous density estimates. For 5-20 km diameters, S types rotate faster and have lower amplitudes than C types. If both populations share the same angular momentum, this may indicate the two types' differing ability to deform under rotational stress. Lastly, we compare our absolute magnitudes (and apparent-magnitude residuals) to those of the Minor Planet Center's nominal (G = 0.15, rotation-neglecting) model; our phase-function plus Fourier-series fitting reduces asteroid photometric RMS scatter by a factor ∼3.