Homogeneous random fractals form a probabilistic extension of self-similar sets with more dependencies than in random recursive constructions. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the Uniform Strong Open Set Condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to zero. Moreover, integral representations are derived for these limits which extend those known in the deterministic case.