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We present a study of even-parity Rydberg exciton states in cuprous oxide using second harmonic generation (SHG) spectroscopy. Excitonic states with principal quantum number n = 5 − 12 were excited by nanosecond pulses around 1143 nm. Using time-resolved single-photon counting, the coherently generated second harmonic was isolated both temporally and spectroscopically from inelastic emission due to lower-lying free and bound excitonic states, which included narrow resonances at 1.993 eV associated with a long lifetime of 641 ± 7 µs. The near transform-limited excitation bandwidth enabled high-resolution measurements of the exciton lineshape and position, from which we obtained values for the quantum defects of the S and D excitonic states associated with the appropriate crystal symmetries. Odd-parity P and F excitonic states were also observed, in accordance with predicted quadrupole-allowed two-photon excitation processes. We compared our measurements to conventional one-photon spectroscopy in the same sample, and find that the SHG spectrum is cut off at a lower principal quantum number (n = 12 vs n = 15). We attribute this effect to a combination of spatial inhomogeneities and local heating, and discuss the prospects for observing higher principal quantum number even-parity states in future experiments.