An extension of the SiameseFWI framework for multi-parameter Elastic Full Waveform Inversion (EFWI) by integrating wavelet transforms, specifically the Haar mother wavelet, to enhance feature extraction from the seismic data. Our Siamese network operates on a self-supervised scheme and consists of two identical convolutional neural network branches with shared weights, which improve data comparison using Euclidean distance as a loss measure. By optimizing the parameters of the Siamese network during the iterative EFWI process and adding a skip connection for stability, we effectively invert the multi-parameter EFWI. We validate our framework with various synthetic and field data examples, demonstrating its enhanced inversion performance compared to traditional EFWI, with minimal additional computational cost.