A parametric study was carried out to investigate the torsional response of structures isolated with the Triple Friction Pendulum (TFP) system with building properties such as uncoupled torsional to lateral frequency ratio, , building slenderness, and mass and stiffness eccentricities. A simplified model of a parametric multistory structure and a small deformation model of the TFP, but with the ability to capture uplift, were used to perform parametric 3D non-linear dynamic analyses. The input ground motions were 20 Chilean records selected using a conditional mean spectrum. Response history results showed that uplift in the devices occurs in structures with large mass eccentricity, and in some extreme cases, normal force values in the TFPs may be as large as 6 times the nominal average gravitational load in each isolator. Torsional envelope response results also show that mass eccentric systems may undergo larger normalized edge amplification factors relative to stiffness eccentric systems, with values 2.5 and 1.8, respectively, and that torsional balance cannot be achieved naturally. Although rotations are small in magnitude due to seismic isolation, displacement values in resisting planes of the superstructure equidistant from the geometric center of the plan will be different. Moreover, structures with large present correlation values between response histories of torsion and translation measured at the geometric center of the plan near one, as well as values above zero for different and plan eccentricity, which implies no torsional control in the weak sense. It is concluded that parameter strongly influences the torsional amplification values computed herein.