The purpose of this article is to elucidate inherent limitations to the performance of parallel MRI. The study focuses on the ultimate signal-to-noise ratio (SNR), which refers to the maximum SNR permitted by the electrodynamics of the signal detection process. Using a spherical model object, it is shown that the behavior of the ultimate SNR imposes distinct limits on the acceleration rate in parallel imaging. For low and moderate acceleration, the ultimate SNR performance is nearly optimal, with geometry factors close to 1. However, for high reduction factors beyond a critical value, the ultimate performance deteriorates rapidly, corresponding to exponential growth of the geometry factor. The transition from optimal to deteriorating performance depends on the electrodynamic characteristics of the detected RF fields. In the near-field regime, i.e., for low B0 and small object size, the critical reduction factor is constant and approximately equal to four for 1D acceleration in the sphere. In the far-field wave regime the critical reduction factor is larger and increases both with B0 and object size. Therefore, it is concluded that parallel techniques hold particular promise for human MR imaging at very high field.