Yes, it is possible to run parallel tests with Browserstack local. You can run upto 100 parallel tests with only one instance Browserstack local without having any performance impact.
Incase, you want to scale even more, you can start another instance of Browserstack local with the flag : --local-identifier <unique_string>
and pass the following capability browserstack.localIdentifier : <unique_string>
in your test. This would ensure the traffic of that test would route through this binary instance with the local identifier. You can run multiple such instances by passing some unique strings.