In this work, a new class of partially coherent Schell-type sources is introduced by modifying its degree of coherence, which is a product of a parabolic function and a Gaussian function. Such sources are confirmed to be physically genuine and may be called parabolic-Gaussian Schell-model (PGSM) sources. The propagating expression of the cross-spectral density function of such PGSM sources is derived in a general linear optical system. In particular, the propagation properties of such PGSM beams in both free space and lens systems are discussed. The results show that such PGSM sources can produce dark-hollow intensity profiles in the regions of free space very near as well as far away from the source plane, or in the focal region of the lens system.