[standevbob]

Yes, we regularly use Stan's MCMC to fit relatively simple time-series regression models or item-response theory type models with 10^5 parameters and 10^6 rows of data on a desktop computer. It can take a day, though. It's much faster with variational inference, but that can be less stable and it doesn't give you the same uncertainty quantification because of the way the KL-divergence is ordered in the objective.

Stan can parallelize multiple chains and it can parallelize the density/gradient calculations in a single chain. But for the latter to be efficient, the chunks being parallelized need to be compute intensive, like you might get in a pharmacometric compartment model where you might have to solve a bunch of differential equations for each of thousands of patients in a clinical trial.