  In this article we define an encoding for parabolic permutations that
distinguishes between parabolic $231$-avoiding permutations. We prove that the
componentwise order on these codes realizes the parabolic Tamari lattice, and
conclude a direct and simple proof that the parabolic Tamari lattice is
isomorphic to a certain $\nu$-Tamari lattice, with an explicit bijection.
Furthermore, we prove that this bijection is closely related to the map
$\Theta$ used when the lattice isomorphism was first proved in (Ceballos, Fang
and M\"uhle, 2020), settling an open problem therein.
