Consider a curve given by y = f(x).
The derivative of the function at any point x = a is given by
f'(a) = `lim_(h->0)(f(a+h) - f(a))/h`
Differentiation is also equivalent to taking two points corresponding to f(a) and f(a+h) and drawing a line between them. [f(a+h)-f(a)]/h is equal to the slope of the line between f(a) and f(a +h) and is the slope of the tangent at x = a as h tends to 0.
For any function, f'(a) either exists if the function is continuous or it does not exist where there are points of discontinuities.
A parabola is a continuous function, with the derivative existing at all points and having a unique value.
Therefore a parabola cannot have more than tangent at any point.