You can send multiple requests simultaneously to the JSON-RPC API, although you should note that some of what happens behind the scenes doesn't run concurrently, so you won't see a full linear speedup.
Also note that multichain-cli is just a wrapper for the JSON-RPC API, so if that runs concurrently without problems, that means the issue is not with the API itself.
So I'm guessing this is some issue with Savoir/Python – hard to go into more detail than that.