Brain tumors rank among the most aggressive and life threatening diseases globally, characterized by rapid progression and high mortality rates. Early and accurate diagnosis is paramount for effective treatment planning and improved patient outcomes. Deep transfer learning has emerged as a powerful approach for classifying the three primary types of brain tumors: glioma, meningioma, and pituitary tumors. However, adapting pretrained models to specific tasks often necessitates sophisticated optimization techniques to achieve optimal performance. In this study, we introduce a novel optimization framework for parameter-based transfer learning in convolutional neural networks (CNNs). Our approach incorporates a Lasso-based regularization term to promote sparsity and control the transferability of parameters. To efficiently solve the resulting optimization problem, we employ the proximal gradient descent method, which is well-suited for handling non-differentiable regularization terms. We validate our method on a three-class brain tumor classification task using a medical MRI dataset, focusing on distinguishing between glioma, meningioma, and pituitary tumors. Comprehensive experiments demonstrate that our approach effectively identifies which parameters to fine-tune and which to freeze, leading to improved classification accuracy. Finally, we compare our method against traditional transfer learning techniques, highlighting its superior performance and practical advantages in medical image classification tasks.