If your pde has spatial locality, parallelization works. I suspect you might be talking about multidimensional diffusions such as occur in finance. These decomp

If your pde has spatial locality, parallelization works. I suspect you might be talking about multidimensional diffusions such as occur in finance. These decompose locally in price space. If the pde is nonlocal there is very little that works in general. I wrote a paper on parallelization in the time dimension, but it works poorly.