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.
It also likely depends on how well conditioned your model is - even if you can get it to run for huge models on reasonable hardware, convergence may not be practical.