  In this paper we establish a scale invariant Harnack inequality for the
fractional powers of parabolic operators $(\partial_t - \mathscr{L})^s$,
$0<s<1$, where $\mathscr{L}$ is the infinitesimal generator of a class of
symmetric semigroups. As a by-product we also obtain a similar result for the
nonlocal operators $(-\mathscr{L})^s$. Our focus is on non-Euclidean
situations.
