This paper investigates the parametric instability behavior of face-gear drives by modeling pinion and face-gear tooth geometry and kinematics. The out-of-plane dynamics of the face-gear are excited by periodically time-varying gear tooth mesh stiffness variations. Here an annular spinning Kirchhoff plate with centrifugal stiffening effects and a moving spring is used to model the face-gear and Floquet theory is employed to determine the system stability. Additionally, this paper considers the maximum bending stresses at the root of pinion tooth to restrict the face-gear size. Tregold's approximation is utilized to calculate the contact-ratio of face-gear drives. Finally, an example of a prototypical helicopter face-gear drive is studied at different operating speeds. The results characterize the parametric instability regions as a function of rotation speed and face-gear disk thickness. Moreover, the minimum critical thickness to sustain the system stability over the entire operating speed region is determined. The analysis and results provide new and important insights into the dynamics and design of lightweight face-gear drives.