Tangent Conics

Suppose we start at the focus common to both the ellipse and the parabola, and shoot a point-like ball at the tangent point $(a,a^2)$ . Because the point bounces off the parabola, it will travel vertically after reflection. However, since it also bounces off the ellipse, it will travel to the ellipse's second focus. Thus, we know that the ellipse's second focus's x-coordinate is equal to $a$ . Thus, $a^2 \le 2014\implies a\le \sqrt{2014}\approx \boxed{44}$ and we are done.