Most states will not have a definite parity, since parity transforms the wave-vector k to -k. Only at special points in the Brillouin zone will definite-parity states exist.
You will have to write down the Schrödinger equation at the symmetry points, and check for each eigenstate. It is not different from what you must do for any other system with parity symmetry. You should find treatments of this in textbooks on condensed matter theory, like the one by Ashcroft and Mermin.
A good way to start is probably to consider the point k=0 (\Gamma point). For other symmetry points you must have k= -k + G, where G is some reciprocal lattice vector.