The, output of a pulsed dye-laser pumped by a frequency-doubled Nd: YAG laser is focused into a heat-pipe oven. The optical wave-mixing processes and parametric oscillations from two-photon excitation of 5d2DJ state in potassium vapor are investigated. Twenty-eight coherent lines within the range of 302-421 nm are generated. Splits and frequency shifts of the parametric oscillations and wavelengths and relative intensities of spectral lines in the wave-mixing processes are explained with a theoretical analysis based on the wave equation, atomic polarization and phase-match condition. The theoretical values are found to be in agreement with the experimental results.