We study the spectral data for the higher direct images of a parabolic Higgs
bundle along a map between a surface and a curve with both vertical and horizontal
parabolic divisors. We describe the cohomology of a parabolic Higgs bundle on a
curve in terms of its spectral data. We also calculate the integral kernel that
reproduces the spectral data for the higher direct images of a parabolic Higgs bundle
on the surface. This research is inspired by and extends the works of Simpson [21]
and Donagi-Pantev-Simpson [7]
