I don't see how 2 distinct parabolas can intersect in more than 2 points.
If you put both parabolas into general form (y=ax^2+bx+c), then you can find the x-values at which they intersect by setting the two right hand sides equal to each other. This is a quadratic equation in x which can have at most 2 solutions.
Edit: I was assuming that the two parabolas were graphs of functions. It is easy to make two parabolas intersect in 3 or 4 points if one has the form y=ax^2+bx+c, and the other has the form x=ay^2+by+c