The study of the poset of hyperbolic structures H(G) on a group G was initiated by Abbott-Balasubramanya-Osin. However, the sub-poset of quasi- parabolic structures is still very far from being understood and several questions remain unanswered. In this talk, I will talk about the motivation behind our work, describe some structural results related to quasi-parabolic structures and thus answer some of the open questions. I will end my talk by discussing ongoing work in the area. This talk contains some joint work with C.Abbott, D.Osin and A.Rasmussen.