We propose and realize truly single-mode, single-cell photonic crystal resonators surrounded by six symmetric waveguides. The waveguides provide different parity-selective loss paths for different resonant cavity modes. We found symmetry conditions where only the hexapole mode is well confined while the other modes are severely leaky along the waveguides. Utilizing the parameters derived from this parity-mismatch principle, the hexapole-mode resonators were fabricated on InP-InGaAsP slabs. Through photopumping, we obtained the hexapole-mode laser operating in a single mode over a wide range of around 100 nm.