  In this paper, we establish a scale invariant Harnack inequality for some
inhomogeneous parabolic equations in a suitable intrinsic geometry dictated by
the nonlinearity. The class of equations that we consider correspond to the
parabolic counterpart of the equations studied by Julin in [10] where a
generalized Harnack inequality was obtained which quantifies the strong maximum
principle. Our version of parabolic Harnack (see Theorem 1.2) when restricted
to the elliptic case is however quite different from that in [10]. The key new
feature of this work is an appropriate modification of the stack of cubes
covering argument which is tailored for the nonlinearity that we consider.
