This paper addresses the local stabilization of constrained nonlinear systems with input saturation described by Takagi-Sugeno fuzzy models with nonlinear consequents. To reduce the design conservativeness, we propose a new delayed multiple-parameterization control approach based on a nonquadratic Lyapunov function with multiple delayed fuzzy summations. Both input saturation and state constraints are explicitly taken into account in the control design procedure. This multiple-parameterization condition is given in terms of linear matrix inequalities. Compared to existing results, the new approach offers a unified and concise control framework to design both non-delayed and delayed multidimensional nonlinear fuzzy controllers. Numerical examples are provided to demonstrate the effectiveness of the proposed multiple-parameterization approach in both reducing the design conservativeness and enlarging the estimation of the domain of attraction.