<p>This paper begins by examining the definition and fundamental properties of the two-dimensional linear canonical wavelet transform (2-D LCWT) within the framework of linear canonical transform theory. The LCWT offers significant advantages in handling multi-scale and multi-directional signals. Building on this, a novel model is proposed based on the parametric Riesz transform and parametric monogenic signal, introducing a parametric monogenic linear canonical wavelet along with its associated transform, referred to as PMLW. Through parametric embedding within the monogenic linear canonical wavelet framework, the proposed transform achieves enhanced flexibility and robustness, facilitating more efficient analysis of intricate features in complex signals. This approach leverages 2-D analytical signal theory to incorporate phase information with directional characteristics, thereby preserving rich directional details in multi-scale analysis. Furthermore, the potential applications of PMLW in image denoising tasks are explored, demonstrating its effectiveness in preserving structural details while suppressing noise.</p>