  In this paper, we establish stability of parabolic Harnack inequalities for
symmetric non-local Dirichlet forms on metric measure spaces under general
volume doubling condition. We obtain their stable equivalent characterizations
in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and
Poincar\'e inequalities. In particular, we establish the connection between
parabolic Harnack inequalities and two-sided heat kernel estimates, as well as
with the H\"older regularity of parabolic functions for symmetric non-local
Dirichlet forms.
