Identifying links within biological networks is important in various biomedical applications. Recent studies have revealed that each node in a network may play a unique role in different links, but most link prediction methods overlook distinctive node roles, hindering the acquisition of effective link representations. Subgraph-based methods have been introduced as solutions but often ignore shared information among subgraphs. To address these limitations, we propose a Subgraph-aware Graph Kernel Neural Network (SubKNet) for link prediction in biological networks. Specifically, SubKNet extracts a subgraph for each node pair and feeds it into a graph kernel neural network, which decomposes each subgraph into a combination of trainable graph filters with diversity regularization for subgraph-aware representation learning. Additionally, node embeddings of the network are extracted as auxiliary information, aiding in distinguishing node pairs that share the same subgraph. Extensive experiments on five biological networks demonstrate that SubKNet outperforms baselines, including methods especially designed for biological networks and methods adapted to various networks. Further investigations confirm that employing graph filters to subgraphs helps to distinguish node roles in different subgraphs, and the inclusion of diversity regularization further enhances its capacity from diverse perspectives, generating effective link representations that contribute to more accurate link prediction.