The forces on a stationary ball are equal, so, where w is the angular velocity, and assuming r' is small,

[1]  mrw² cos t = mg sin t

Or,

[2]  tan t = (rw²)/g

The general form of a parabola, where k is a constant, is

[3]  z = kr²

And, taking the derivative, which is also equal to the tangent of t,

[4]  dz/dr = 2kr = tan t

Substituting equation [4] into equation [2] gives

[5]  w = (2kg)^½


But, if the parabola is anything like [bris] shows in her drawings, you really do have to take the ball diameter into account. The more exact equation, where d is the diameter of the balls, is,

[6]  w = (2kg)^½ * (r/(r-(d/2 sin t)))^½

So, a perfect parabola cannot be used; the surface has to be adjusted according to ball diameter.