Speckle noise severely limits the quality of images acquired from coherent imaging systems such as Synthetic Aperture Radar (SAR) and medical ultrasound. Traditional second-order PDE-based despeckling approaches, although popular, often introduce staircase artifacts and blur fine details. To overcome these limitations, we present a nonlinear, fourth-order coupled hyperbolic-parabolic PDE model that effectively reduces noise while preserving the structure. The framework consists of two evolution equations: one governing fourth-order diffusion for effective speckle reduction and smooth intensity transitions, and another refining an edge indicator to protect textures and structural features. The diffusion coefficient is adaptively constructed using both the image intensity variable u and a grayscale-based indicator function, ensuring structure-aware denoising while avoiding blocky artifacts and preserving fine details. We also prove the existence of a weak solution to the proposed model by applying Schauder fixed-point theorem. A finite-difference scheme with Gauss Seidel iteration is employed for efficient implementation. We compare the proposed model with the existing coupled second-order PDE model (HPCPDE) and the fourth-order telegraph diffusion model (TDFM). The results show that our model consistently outperforms these approaches. Experiments on standard grayscale images, real SAR and ultrasound data, as well as speckle-corrupted color images, demonstrate that the proposed method achieves superior performance over conventional PDE-based techniques in terms of PSNR, MSSIM, and Speckle Index.