  We use the auxiliary-fields Monte Carlo method for the shell model in the
complete $(pf+0g_{9/2})$-shell to calculate level densities. We introduce
parity projection techniques which enable us to calculate the parity dependence
of the level density. Results are presented for $^{56}$Fe, where the calculated
total level density is found to be in remarkable agreement with the
experimental level density. The parity-projected densities are well described
by a backshifted Bethe formula, but with significant dependence of the
single-particle level-density and backshift parameters on parity. We compare
our exact results with those of the thermal Hartree-Fock approximation.
