  As a classical object, the Tamari lattice has many generalizations, including
$\nu$-Tamari lattices and parabolic Tamari lattices. In this article, we unify
these generalizations in a bijective fashion. We first prove that parabolic
Tamari lattices are isomorphic to $\nu$-Tamari lattices for bounce paths $\nu$.
We then introduce a new combinatorial object called `left-aligned colorable
tree', and show that it provides a bijective bridge between various parabolic
Catalan objects and certain nested pairs of Dyck paths. As a consequence, we
prove the Steep-Bounce Conjecture using a generalization of the famous zeta map
in $q,t$-Catalan combinatorics. A generalization of the zeta map on parking
functions, which arises in the theory of diagonal harmonics, is also obtained
as a labeled version of our bijection.
