  Using variational methods, we prove local higher integrability for the
minimal p-weak upper gradients of parabolic quasiminimizers in metric measure
spaces. We assume the measure to be doubling and the underlying space to be
such that a weak Poincar\'e inequality is supported. We give proofs to density
results concerning the space of test functions used when proving estimates for
parabolic quasiminimizers.
