Self-supervised deep learning-based methods have shown great promise for accelerated magnetic resonance imaging (MRI) reconstruction, achieving high image quality without requiring fully sampled data for training. These methods typically partition the acquired data into two disjoint subsets to construct input-target pairs for optimizing the reconstruction network. However, existing approaches perform this partition exclusively within the spatial frequency (k-space) domain, leaving the coil dimension unexplored. To enforce full exploitation of signal correlation across receiver coils, we propose CoilDrop-MRI, which applies coil-wise dropout to the input and uses the dropped data as training targets in a self-supervised framework. This method is integrated into unrolled architectures in both image-domain (SENSE) and k-space (SPIRiT) formulations. We further demonstrate its versatility by extending CoilDrop-MRI to multi-shot, phase-corrected diffusion MRI (dMRI) reconstruction. CoilDrop-MRI is extensively validated on multi-site, multi-field-strength (0.3T, 0.55T, and 3T), and multi-modality (T1-weighted, T2-weighted, T2-FLAIR, and dMRI) datasets and consistently outperforms state-of-the-art self-supervised methods, achieving quality comparable to supervised reconstruction methods without requiring fully sampled reference training data. Moreover, CoilDrop-MRI exhibits strong data efficiency and robust generalization across imaging conditions, establishing it as a practical and versatile framework for self-supervised parallel MRI reconstruction.