In an async function, the only guarantee you have is that code between two await points is not interruptible. As you have more and more async operations, this guarantee gets pretty weak (if you have more than one pending async task and the current task yields, which one will run next?) and you have to start employing the usual locking mechanisms to ensure correctness.
Highly concurrent code need not execute in parallel. Concurrency enables parallelism; concurrency does NOT mandate parallelism.
On top of that, it actually is possible in JavaScript environments like Node.js to write code that can, in fact, run in parallel.