The closest I've seen to anything that actually does that is UML. But it does so only by having multiple different diagrams. That's like having a drawing with a front view and a side view - it's not a 3D diagram, it's two 2D diagrams that you can use to kind of see what's going on in 3D.
I don't know of any good system of notation that does what you're asking, nor any software for drawing or viewing such diagrams. But I agree that 2D flowcharts don't represent parallelism well, and that 3D looks like a better answer.
UML is non-rigorous and has a whole lot of pointless incidental complexity. Intuitive representations of parallelism are fairly natural in data flow, but that is precisely dual to flowcharts; you can not represent both on the same 2D diagram without some very precise rules about what portions of the diagram are intended to represent each. Proof nets give you something not unlike this.
Oh, I never said UML was ideal. I merely said that, if you squint, you could see it as a clumsy, inadequate, and probably unintentional half-step in the direction of crimsonalucard's idea.
The closest I've seen to anything that actually does that is UML. But it does so only by having multiple different diagrams. That's like having a drawing with a front view and a side view - it's not a 3D diagram, it's two 2D diagrams that you can use to kind of see what's going on in 3D.
I don't know of any good system of notation that does what you're asking, nor any software for drawing or viewing such diagrams. But I agree that 2D flowcharts don't represent parallelism well, and that 3D looks like a better answer.