Recent advances in pre-trained vision transformers have shown promise in parameter-efficient audio-visual learning without audio pre-training. However, few studies have investigated effective methods for aligning multimodal features in parameter-efficient audio-visual transformers. In this paper, we propose MA-AVT, a new parameter-efficient audio-visual transformer employing deep modality alignment for corresponding multimodal semantic features. Specifically, we introduce joint unimodal and multimodal token learning for aligning the two modalities with a frozen modality-shared transformer. This allows the model to learn separate representations for each modality, while also attending to the cross-modal relationships between them. In addition, unlike prior work that only aligns coarse features from the output of unimodal encoders, we introduce blockwise contrastive learning to align coarse-to-fine-grain hierarchical features throughout the encoding phase. Furthermore, to suppress the background features in each modality from foreground matched audio-visual features, we introduce a robust discriminative foreground mining scheme. Through extensive experiments on benchmark AVE, VGGSound, and CREMA-D datasets, we achieve considerable performance improvements over SOTA methods.