Assessing drought on the Tibetan Plateau (TP) is essential for understanding the spatial characteristics of the region's hydrological regimes, as well as the trends of the evolving climate-ecological system. Based on the precipitation and the potential evapotranspiration data from the ERA5-Land dataset, this study evaluated the drought characteristics on the TP from 1950 to 2022 using the Standardised Precipitation Evapotranspiration Index (SPEI) and the Run theory. The analysis of SPEI at multiple time scales—SPEI-1 (monthly), SPEI-3 (seasonal), and SPEI-12 (annual)—consistently revealed a pronounced drying trend across the TP over the past 70 years. Notably, the wet-to-dry transition points were detected in 2000 and 2001 based on the SPEI-1 and SPEI-3 series, respectively, while the SPEI-12 series indicated a later shift occurring in 2004. In addition, the SPEI values indicated a spatial trend of increasingly severe drought conditions from south to north. Particularly, pronounced drought stress was observed in the Brahmaputra, Ganges, and Indus basins, as well as in the southern regions of the Inner Plateau. Furthermore, the relationships between drought variability and key climatic drivers—including sunspots, El Niño-Southern Oscillation (ENSO), temperature (TEMP), and annual mean areal precipitation (AP)—were identified through cross-wavelet transform (XWT) and wavelet coherence analysis (WTC). In general, the analysis showed that the SPEI values on the TP were negatively correlated to ENSO and TEMP but positively correlated to sunspots and AP. This study offers a comprehensive spatiotemporal assessment of drought dynamics across the TP, providing critical insights for regional water resource management and climate change adaptation planning.