Exponential analogue of NC?
Time in circuits corresponds to the depth.
Therefore by polynomial time means polynomial depth.

The number of processors is the size of the circuit, 
i.e. number of the gates in the circuit.
So by exponential number of processors you allow exponential size.
This would be the class $\mathsf{DepthSize}(n^{O(1)}, 2^{n^{O(1)}})$.
But every function is already in $\mathsf{DepthSize}(2, 2^{n^{O(1)}})$
(think of the CNF of the function you want to compute).

The take away is that exponential number of processors is 
too strong to be useful by itself.

One reasonable restriction to put is to limit the amount of communication
between different processes.
E.g. we each process can only communicate with 
only polynomially many other processes and the messages have polynomial size.
That would be $\mathsf{PSpace}$ as explained in answers to Aterm's question on cstheory.
Another way to see it to remember that 
$\mathsf{PSpace} = \mathsf{ATime}(n^{O(1)})$,
problems computable by alternating Turing machines in polynomial time.
Alternation in Turing machines is essentially forking new processes and then joining after they finish by taking the conjunction/disjunction of their return values.