We study the cohomological equation for discrete horocycle maps on \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} and \begin{document}$ {\rm SL}(2, \mathbb{R})\times {\rm SL}(2, \mathbb{R}) $\end{document} via representation theory. Specifically, we prove Hilbert Sobolev non-tame estimates for solutions of the cohomological equation of horocycle maps in representations of \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} . Our estimates improve on previous results and are sharp up to a fixed, finite loss of regularity. Moreover, they are tame on a co-dimension one subspace of \begin{document}$ \operatorname{\mathfrak s\mathfrak l}(2, \mathbb{R}) $\end{document} , and we prove tame cocycle rigidity for some two-parameter discrete actions, improving on a previous result. Our estimates on the cohomological equation of horocycle maps overcome difficulties in previous papers by working in a more suitable model for \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} in which all cases of irreducible, unitary representations of \begin{document}$ {\rm SL}(2, \mathbb{R}) $\end{document} can be studied simultaneously. Finally, our results combine with those of a very recent paper by the authors to give cohomology results for discrete parabolic actions in regular representations of some general classes of simple Lie groups, providing a fundamental step toward proving differential local rigidity of parabolic actions in this general setting.