Training memory-augmented Temporal Graph Neural Networks (M-TGNNs) efficiently and accurately remains challenging due to memory staleness, which arises when temporally dependent events are processed in the same batch and severely degrades accuracy at large batch sizes. We introduce PRISM, an M-TGNN training system that achieves staleness-freedom without giving up GPU parallelism by using multi-versioned memory vectors, so that each event in a batch can consume the memory version that is temporally consistent for it. PRISM formalizes a relaxed notion of staleness-freedom called lazy freshness, which allows for more parallelism than existing staleness-free approaches, and implements it through a multi-versioned memory refinement algorithm over a lightweight memory computation graph. On five temporal-graph benchmarks and three M-TGNN models (TGN, TNCN, APAN), PRISM improves the accuracy of existing models by up to 28% and surpasses the TGB leaderboard by 9.2%, while keeping training time competitive with parallel stale-memory systems (TGL, ETC) and consistently lower than stricter staleness-free baselines. PRISM thus provides a practical, staleness-free foundation for temporal graph learning.