Hi,
From your description I cannot comment on what’s happening, and whether there are any errors or inefficiencies with what you are doing, but if you are using ODEs in principle there is not much to parallelize in a in-practice markovian system like the SIR. I am guessing the step size optimization could be parallelized, not sure if it would be very efficient. There has been some discussion here about the possibility of using Discrete Time Solver instead of ODEs to get rid of the step size optimization cost, but there are caveats and to me it’s not clear that the benefits would be considerable.
I’m not familiar with the PETSc library, maybe others here are, but providing additional details on what you expect to accomplish with your scheme and how any tools external to Stan are doing it it would make it more understandable to most people here.