  This paper studies the geometry and combinatorics of three interrelated
varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg
varieties. We prove that each parabolic Hessenberg variety is the pullback of a
Steinberg variety under the projection of the flag variety to an appropriate
partial flag variety and we give three applications of this result. The first
application constructs an explicit paving of all Steinberg varieties in Lie
type $A$ in terms of semistandard tableaux. As a result, we obtain an
elementary proof of a theorem of Steinberg and Shimomura that the well-known
Kostka numbers count the maximal-dimensional irreducible components of
Steinberg varieties. The second application proves an open conjecture for
certain parabolic Hessenberg varieties in Lie type A by showing that their
Betti numbers equal those of a specific union of Schubert varieties. The third
application proves that the irreducible components of parabolic Hessenberg
varieties are in bijection with the irreducible components of the Steinberg
variety. All three of these applications extend our geometric understanding of
the three varieties at the heart of this paper, a full understanding of which
is unknown even for Springer varieties, despite over forty years' worth of
work.
