The central density profiles in dwarf galaxy halos depend strongly on the nature of dark matter (DM). Recently, in Malhan et al. we employed N-body simulations to show that the cuspy cold DM subhalos predicted by cosmological simulations can be differentiated from cored subhalos using the properties of accreted globular cluster (GC) streams since these GCs experience tidal stripping within their parent halos prior to accretion onto the Milky Way. We previously found that clusters that are accreted within cuspy subhalos produce streams with larger physical widths and higher dispersions in line-of-sight velocity and angular momentum than streams that are accreted within cored subhalos. Here, we use the same suite of simulations to demonstrate that the dispersion in the tangential velocities of streams ( ${\sigma }_{{v}_{\mathrm{Tan}}}$ ) is also sensitive to the central DM density profiles of their parent dwarfs and GCs that they were accreted from; cuspy subhalos produce streams with larger ${\sigma }_{{v}_{\mathrm{Tan}}}$ than those accreted inside cored subhalos. Using Gaia EDR3 observations of multiple GC streams we compare their ${\sigma }_{{v}_{\mathrm{Tan}}}$ values with simulations. The measured ${\sigma }_{{v}_{\mathrm{Tan}}}$ values are consistent with both an "in situ" origin and with accretion inside cored subhalos of M ~ 10^8-9 M _⊙ (or very low-mass cuspy subhalos of mass ~10⁸ M _⊙). Despite the large current uncertainties in ${\sigma }_{{v}_{\mathrm{Tan}}}$ , we find a low probability that any of the progenitor GCs were accreted from cuspy subhalos of M ≳ 10 ⁹ M _⊙. The uncertainties on Gaia tangential velocity measurements are expected to decrease in future and will allow for stronger constraints on subhalo DM density profiles.