The ability of Graph Neural Networks (GNNs) to capture long-range and global topology information is limited by the scope of conventional graph Laplacian, leading to unsatisfactory performance on some datasets, particularly on heterophilic graphs. To address this limitation, we propose a new class of parameterized Laplacian matrices, which provably offers more flexibility in controlling the diffusion distance between nodes than the conventional graph Laplacian, allowing long-range information to be adaptively captured through diffusion on graph. Specifically, we first prove that the diffusion distance and spectral distance on graph have an order-preserving relationship. With this result, we demonstrate that the parameterized Laplacian can accelerate the diffusion of long-range information, and the parameters in the Laplacian enable flexibility of the diffusion scopes. Based on the theoretical results, we propose topology-guided rewiring mechanism to capture helpful long-range neighborhood information for heterophilic graphs. With this mechanism and the new Laplacian, we propose two GNNs with flexible diffusion scopes: namely the Parameterized Diffusion based Graph Convolutional Networks (PD-GCN) and Graph Attention Networks (PD-GAT). Synthetic experiments reveal the high correlations between the parameters of the new Laplacian and the performance of parameterized GNNs under various graph homophily levels, which verifies that our new proposed GNNs indeed have the ability to adjust the parameters to adaptively capture the global information for different levels of heterophilic graphs. They also outperform the state-of-the-art (SOTA) models on 6 out of 7 real-world benchmark datasets, which further confirms their superiority.