As we've pretty much got the answer to the original question I'm going to take a moment to quickly (and basically) cover a thing
Good clarification. Makes sense that even if the GPU has lots of pixel/vertex/computing units, the system controlling them isn't necessarily as parallel-friendly. For a non-hw person the number three sounds like a curious choice, but in any case it seems to make some intuitive sense to have the number close to a common number of CPU cores. That's excluding hyper-threading but that's an Intel thing so doesn't matter to folks at AMD. (Though there's the consoles with more cores...)
So, the number '3' has nothing to do with CPU core counts; when it comes to GPU/CPU reasoning very little of one directly impacts the other.
A GPU works by consuming 'command packets'; the OpenGL calls you make get translated by the driver into bytes the GPU can natively read and understand, in the same way a compiler transforms your code to binary for the CPU.
The OpenGL and D3D11 model of a GPU presents a case where the command stream is handled by a single 'command processor' which is the hardware which decodes the command packets to make the GPU do it's work. For a long time this was probably the case too so the conceptual model 'works'.
However, a recent GPU, such as AMD's Graphics Core Next series is a bit more complicated than that as the interface which deals with the commands isn't a single block but in fact 3 which can each consume a stream of commands.
First is the 'graphics command processor'; this can dispatch graphics and compute workloads to the GPU hardware to work - glDraw/glDispatch family of functions - and is where your commands end up.
Secondly there is the 'compute command processors' - these can handle compute only workloads. Not exposed via GL, I think OpenCL can kind of expose them but with Mantle it is a separate command queue. (The driver might make use of them as well behind the scenes)
Finally 'dma commands' which is a separate command queue to move data to/from the GPU which is handled in OpenGL behind the scenes by the driver (but in Mantle would allow you to kick your own uploads/downloads as required.
So the command queues as exposed by Mantle more closely mirror the operation of the hardware (it still hides some details) which explains why you have three, to cover the 3 types of command work the GPU can do.
If you are interested AMD have made a lot of this detail available which is pretty cool.
(Annoyingly NV are very conservative about their hardware details which makes me sad
To be clear, you don't need to know this stuff although I personally find it interesting - this is also a pretty high level overview of the situation so don't take it as a "this is how GPUs work!" kinda thing