The moduli spaces of parabolic connections are referred to as phase spaces of the differential equations determined by the isomonodromic deformations. In this talk, we extend these moduli spaces by considering the moduli spaces of parabolic connections with a quadratic differential. We show that the extended moduli spaces are equipped with the structure of twisted cotangent bundles. This extension of the moduli spaces is related to the classical trick of turning a time dependent Hamiltonian flow into an autonomous one by adding variables.