  We study parabolic quasi-variational inequalities (QVIs) of obstacle type.
Under appropriate assumptions on the obstacle mapping, we prove the existence
of solutions of such QVIs by two methods: one by time discretisation through
elliptic QVIs and the second by iteration through parabolic variational
inequalities (VIs). Using these results, we show the directional
differentiability (in a certain sense) of the solution map which takes the
source term of a parabolic QVI into the set of solutions, and we relate this
result to the contingent derivative of the aforementioned map. We finish with
an example where the obstacle mapping is given by the inverse of a parabolic
differential operator.
