Are you assuming that the axis is vertical? If so then any such parabola can be written as [itex]y= ax^2+ bx+ c[/itex] and you need three equations to solve for the three coefficients, a, b, c. You are given two points, [itex](x_1,y_1)[/itex] and [itex](x_2,y_2)[/itex] on the parabola so [itex]y1= ax_1^2+ bx_1+ c[/itex] and [itex]y2= ax_2^2+ bx_2+ c[/itex]. Those are two of the equations. You also know the "length" of the parabola between [itex]x_1[/itex] and [itex]x_2[/itex]- it's the "time of travel" divided by the uniform velocity (I assume you know that velocity- otherwise you do not have enough information to determine the parabola). Write out the equation for the arc-length of [itex]y= ax^2+ bx+ c[/itex] and set it equal to that length. That gives you a third equation for a, b, and c.