In the general case a spherically symmetric potential, the Time Independent Schrodinger Equation is separable in spherical coordinates but not in cartesian, or cylindrical coordinate as in general $V(r)\neq{V(x)+V(y)+V(z)}$, and $V(r)\neq{V(\rho)+V(z)}$. In the special case of a spherically symmetric parabolic potential, $V(r)=ar^2=a\sqrt{x^2+y^2+z^2}^2=a\left(x^2+y^2+z^2\right)$ and so $V(r)=V(x)+V(y)+V(z)$ meaning that the Time Independent Schrodinger Equation $\nabla^2\Psi+(E-V)\Psi=0$ is separable in cartesian coordinates. Also $\rho=\sqrt{x^2+y^2}$ meaning that $\rho^2=\sqrt{x^2+y^2}^2=x^2+y^2$, meaning that $a\left(x^2+y^2+z^2\right)=a\left(\rho^2+z^2\right)$, which means that $V(r)=V(\rho)+V(z)$, so the Time Independent Schrodinger Equation is also separable in cylindrical coordinates in the case of a spherically symmetric parabolic potential.

Does this mean that for a spherically symmetric parabolic potential, in addition to using separation of variables in spherical coordinates, we also need to use separation of variables for cartesian in order to find all the bound states? If not why don't we need to use separation of variables in cartesian and cylindrical coordinates in order to find all bound states for a spherically symmetric parabolic potential?