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The steepness of the central density profiles of dark matter (DM) in low-mass galaxy halos (e.g. dwarf galaxies) is a powerful probe of the nature of DM. We propose a novel scheme to probe the inner profiles of galaxy subhalos using stellar streams. We show that the present day morphological and dynamical properties of accreted globular cluster (GC) streams - those produced from tidal stripping of GCs that initially evolved within satellite galaxies and later merged with the Milky Way (MW) - are sensitive to the central DM density profile and mass of their parent satellites. GCs that accrete within cuspy CDM subhalos produce streams that are physically wider and dynamically hotter than streams that accrete inside cored subhalos. A first comparison of MW streams "GD-1" and "Jhelum" (likely of accreted GC origin) with our simulations indicates a preference for cored subhalos. If these results hold up in future data, the implication is that either the DM cusps were erased by baryonic feedback, or their subhalos naturally possessed cored density profiles implying DM models beyond CDM. Moreover, accreted GC streams are highly structured and exhibit complex morphological features (e.g., parallel structures and "spurs"). This implies that the accretion scenario can naturally explain the recently observed peculiarities in some of the MW streams. We also propose a novel mechanism for forming "gaps" in streams when the remnant of the parent subhalo later passes through the stream. This encounter can last a longer time (and have more of an impact) than the random encounters with DM subhalos previously considered, because the GC stream and its parent subhalo are on similar orbits with small relative velocities. Current and future surveys of the MW halo will uncover numerous faint stellar streams and provide the data needed to substantiate our preliminary tests with this new probe of DM.