Uncertainty quantification (UQ) in eruption source parameters, like tephra volume, plume height, and umbrella cloud radius, is a challenge for volcano scientists because tephra deposits are often sparsely sampled due to burial, erosion, and related factors. We find that UQ is improved by coupling an advection-diffusion model with two Bayesian inversion approaches: (a) a robust but computationally expensive Generalized Likelihood Uncertainty Estimation algorithm, and (b) a more approximate but inexpensive parameter estimation algorithm combined with first-order, second-moment uncertainty estimation. We apply the two inversion methods to one sparsely sampled tephra fall unit from the 2070 BP El Misti (Peru) eruption and obtain: Tephra mass 0.78–1.4 × 1011 kg; umbrella cloud radius 4.5–16.5 km, and plume height 8–35 km (95% confidence intervals). These broad ranges demonstrate the significance of UQ for eruption classification based on mapped deposits, which has implications for hazard management.