A conjecture of Menasco and Reid states that a hyperbolic knot complement does not contain a closed embedded totally geodesic surface. One heuristic that is used to study the absence of such a submanifold is parabolic cohomology–in particular, if the parabolic cohomology is known to vanish in specific settings, then it serves as an obstruction to bending along a totally geodesic hypersurface. In this talk, we consider the Borromean rings complement, which is known to not admit any closed embedded totally geodesic surfaces but still has interesting parabolic cohomology. We construct a complex of surfaces that can be used to explain these deformations.