Given a PRAM may use arbitrarily many processors, why is Hamiltonian Cycle not in NC?
If I give all of the permutations of the vertices of a graph as input


That's not the input for Hamiltonian Cycle; you only get the graph.


  then I can assign a processor to each permutation


Since there are $n! \in \omega(c^k)$ many permutations, you don't have that many processors.

Here's the twist: if you were allowed to use all permutations as input, you'd of course have polynomially many processors in $n!$ at your disposal. But since your input is the graph you only get polynomially many in $n$.