[en] We present an investigation of resonances with natural and unnatural parities in the positron-potassium system using the complex scaling method. A model potential is used to represent the interaction between the core and the valence electron. Explicitly correlated Gaussian wave functions are used to represent the correlation effects between the valence electron, the positron and the K⁺ core. Resonance energies and widths for two partial waves (S- and P-wave) below the $\mathrm{K}(4p,5s,5p,4d,4f)$ excitation thresholds and positronium n = 2 formation threshold are calculated for natural parity. Resonance states for P ^e below the $\mathrm{K}(4d)$ excitation threshold and positronium n = 2, 3 formation thresholds are calculated for unnatural parity which has not been previously reported. Below both positronium thresholds we have found a dipole series of resonances, with binding energies scaling in good agreement with exceptions from an analytical calculation. The present results are compared with those in the literature. (paper)