Assuming that the parabola has vertical axis, then the equation can be written y= ax^2+ bx+ c. You need three conditions to solve for the three coefficients, a, b, and c.
Saying that the given point (x_0,y_0) is on the parabola means that y_0= ax_0^2+ bx_0+ c. Saying that the given point (x_1,y_1) is on the parabola means that y_1= ax_1^2+ bx_1+ c. That gives you two of the three equations you need.
If y= ax^2+ bx+ c, then y'= 2ax+ b so the arc-length between x_0 and x_1 is given by
\int_{x_0}^{x_1}\sqrt{1- (2ax+ b)^2}dx
If you know the arc-length between the two points, you would have the third. Calculate the formula for arclength between the two points and set equal to that. Of course, if you know the "uniform velocity", the arclength is just the time divided by that velocity. If you don't know the velocity, then, as mathman said, you don't have enough information to determine the parabola.
