The problem that we had with the spherical lens was that the rays furthest from the center did not focus at the same point as the central rays. Instead of a spherical surface one could instead consider a parabolic surface which is not as curved further from the center. A similar calculation for parallel rays passing through a parabolic lens with similar curvature and thickness at its center as we had for the spherical lens shows that the parabolic lens has a better focus. Again the blue dot marks the center of curvature for the left side of the lens and the cyan dot marks the focal point calculated by the lensmaker's equation.
Enlargement of the region about the focus shows that there is still some aberration but the convergence of the rays is better.
Again the difference in position between the center of curvature and the focal point is due to the thickness of the lens. One gets the same converging power for a parabolic lens with the same central curvature. The fact that the outer rays converge too soon tells us that the parabolic lens still has too much curvature as we move away from the center. It would take at least a 4th degree surface to get a better focus than one gets with a parabolic lens.