Potentially dumb question: If pure functional languages are "inherently parallel", why aren't the programs I write in them magically faster?

Context: I still see talks & slide decks claiming that we should be using (pure) FP because it's "inherently parallel" but I don't know of much evidence about this.

I can reasonably espouse FP for many other reasons but performance isn't one of them.

One example is from @emeryberger's research: He found that even though the programming model could support parallel execution, the memory allocators of the time weren't all that good at parallel allocation.

So maybe it's the fault of the supporting runtime systems?