Phase control of parametric modulation in coupled oscillator networks enables these networks to access dynamical states with desired spatiotemporal symmetries. Floquet theory-based approaches successfully predict such states in linear systems, but whether their symmetry properties persist under nonlinearity remains largely unexplored. Here, we establish the existence of nonlinear chiral steady states in a ring of three coupled parametric oscillators with modulation phases chosen to selectively amplify a circulating mode in the linearized system. We find that a cubic nonlinearity limits the exponential growth of the amplified mode, producing a steady finite-amplitude motion that retains the expected chirality. By exploiting the system's spatiotemporal symmetry, we reduce the dynamics to a single averaged equation that quantitatively predicts nonlinear trajectories, steady-state amplitudes, and characteristic time scales of the full three-body system. The chiral steady states possess finite basins of attraction and are accessible from wide ranges of initial conditions and system parameters. Finite-element simulations of elastic plate resonators quantitatively reproduce these features, establishing the relevance of the reduced model to realistic continuum systems. Our results demonstrate that desirable properties of linear time-modulated systems, such as chirality and directional amplification, persist into strongly nonlinear regimes, opening pathways to robust nonreciprocal signal routing and amplification in parametrically driven platforms.