We study dense nuclear matter and the chiral phase transition in a SU(2) parity doublet model at zero temperature. The model is defined by adding the chiral partner of the nucleon, the N, to the linear sigma model, treating the mass of the N as an unknown free parameter. The parity doublet model gives a reasonable description of the properties of cold nuclear matter, and avoids unphysical behaviour present in the standard SU(2) linear sigma model. If the N is identified as the N(1535), the parity doublet model shows a first order phase transition to a chirally restored phase at large densities, $rho approx 10 rho_0$, defining the transition by the degeneracy of the masses of the nucleon and the N. If the mass of the N is chosen to be 1.2 GeV, then the critical density of the chiral phase transition is lowered to three times normal nuclear matter density, and for physical values of the pion mass, the first order transition turns into a smooth crossover.
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