Parametric feedback control is essential for optimizing and stabilizing a dynamical system by minimizing a predefined cost functional. Despite its proven effectiveness, the applicability of feedback control is often limited by the high dimensionality of the discretized problem, especially in parametric settings. To address these challenges, we apply Dynamical Low-Rank Approximation (DLRA). In practice, we built a set of basis functions, spanning a reduced space, which evolves in time following the prescribed dynamical system. DLRA accelarates the solution of the State-Dependent Riccati Equations (SDREs), yielding efficient and accurate solutions for high-dimensional feedback control problems. To increase the accuracy of the approach, we propose Riccati-based DLRA (R-DLRA). Namely, we enrich the DLRA basis with information related to the solution of the SDRE. To further increase the acceleration of the process, we propose a Cascade Newton-Kleinman (C-NK) algorithm for the SDRE, which leverages prior parametric and time knowledge of the Riccati solution, to improve the convergence of Newton-based methods. Our approach provides fast and accurate solutions for infinite horizon optimal control by constructing a low-dimensional representation of the evolving system, enhancing both accuracy and real-time control across multiple parametric instances. The proposed R-DLRA approach outperforms the standard DLRA, global Proper Orthogonal Decomposition (POD), and Riccati-based POD both in terms of accuracy with respect to the high fidelity model and computational costs.