  We construct an action of the graded double affine Hecke algebra (DAHA) on
the parabolic Hitchin complex, extending the affine Weyl group action
constructed in \cite{GSI}. In particular, we get representations of the
degenerate DAHA on the cohomology of parabolic Hitchin fibers. We also
generalize our construction to {\em parahoric} versions of Hitchin stacks,
including the construction of 'tHooft operators as a special case. We then
study the interaction of the DAHA action and the cap product action given by
the Picard stack acting on the parabolic Hitchin stack.
