[?] Determine start and end points of parabolic arc, given points along arc

Given a 3D parabolic arc starting from [X0, Y0, 0] and ending at [X1, Y1, 0], where X0/X1/Y0/Y1 are unknown, how can I determine the end points of the arc if I have the following information?

• Multiple points ([Xn, Yn, Zn], [Xn+1, Yn+1, Zn+1], [Xn+2, Yn+2, Zn+2], ...) along the path of the arc
• Maximum Z value of the arc at its midpoint is governed by the function [;f(X, Y) = 96 + 3\sqrt{(X_1 - X_0)^2 + (Y_1 - Y_0)^2};]
• The "start" and "End" points are assumed to be where Z=0
• Z can never be negative