abstract: We consider 3-dimensional parabolic flows which are renormalised by circle extensions of Anosov diffeormorphisms (i.e., partially hyperbolic). This class of flows includes nilflows on the Heisenberg nilmanifold which are renormalised by partially hyperbolic automorphisms. The transfer operators associated to these partially hyperbolic maps have been studied by Faure \& Tsujii (Ast\'erisque v.375, 2015), as operators acting on anisotropic Hilbert spaces. We use this spectral information to describe the deviation of ergodic averages and solutions of the cohomological equation for the parabolic flow. The method used doesn't rely on representation theory and so gives the potential to extend beyond the classes where results are already known. Joint work with Lucia D. Simonelli.