Photoluminescence measurements at 5 K on wafers containing parabolic quantum wells fabricated by molecular-beam expitaxy with the GaAs-${\mathrm{Al}}_{0.3}$${\mathrm{Ga}}_{0.7}$As system reflect harmonic oscillator-like electron and hole levels. The many observed heavy-hole transitions can be fitted accurately with a model that divides the energy-gap discontinuity ${\ensuremath{\Delta}E}_{g}$ equally between the conduction and valence-band wells. This is in marked contrast to the usual ${\ensuremath{\Delta}E}_{c}=0.85{\ensuremath{\Delta}E}_{g}$ and ${\ensuremath{\Delta}E}_{v}=0.15{\ensuremath{\Delta}E}_{g}$ generally assumed for square wells. Experiment and theory show that parabolic wells can lead to parity-allowed $\ensuremath{\Delta}n=2$ ("forbidden") transitions with strengths greater than that of nearby $\ensuremath{\Delta}n=0$ ("allowed") transitions.