Hopefully, someone younger than myself will remember the formulas which defines a parabola and an elipse, but that's too far back for me. What I do know about parabolic springs is that it is their construction that makes them different than old fashioned leaf springs. You can see that the center is the thickest cross sectional area and they become progressively thiner towards the ends, so their first reactions are at the ends and increases in deflection are resisted by increases in section thickness. I think the X-Y plot of force and deflection is parabolic, not the shape of the spring.
Old fashioned leaf springs were made up of pieces of flat, equal thickness sections of high carbon steel. The more resistance to force they needed, the more leaves they used in their construction. Both types are progressive in delections but the old leaf springs are very stepped in their progression and parabolics are very smooth, because they are a single homogeneous piece of metal
As far back as the late 70s, Ford used Parabolic leaf springs in their pickup trucks.