1. Symmetry and Circle Equation:
By symmetry, the center of the circle must lie on the y-axis, i.e., $x = 0$. Therefore, the equation of the circle can be written as:
$$x^2 + (y - n)^2 = 49$$
where $n$ is the y-coordinate of the center of the circle.

2. Intersection Points:
The circle touches the parabola $y = x^2$ at two points, which by symmetry are $(a, b)$ and $(-a, b)$. Substituting $y = x^2$ into the circle's equation, we get:
$$x^2 + (x^2 - n)^2 = 49$$
Simplifying, we have:
$$x^2 + (x^4 - 2nx^2 + n^2) = 49$$
$$x^4 - x^2(2n - 1) + n^2 - 49 = 0$$

3. **Quadratic in $y$**:
Let $y = x^2$. Then the equation becomes:
$$y + (y - n)^2 = 49$$
$$y + y^2 - 2ny + n^2 = 49$$
$$y^2 + (1 - 2n)y + n^2 - 49 = 0$$

4. Discriminant Condition:
Since the circle touches the parabola at exactly two points with the same y-coordinate, the quadratic equation in $y$ must have exactly one solution. Therefore, the discriminant of the quadratic equation must be zero:
$$\Delta = (1 - 2n)^2 - 4(n^2 - 49) = 0$$
Simplifying the discriminant:
$$(1 - 2n)^2 - 4(n^2 - 49) = 0$$
$$1 - 4n + 4n^2 - 4n^2 + 196 = 0$$
$$1 - 4n + 196 = 0$$
$$197 - 4n = 0$$
$$4n = 197$$
$$n = \frac{197}{4}$$

5. Conclusion:
Thus, the center of the circle is at:
$$(0, \frac{197}{4})$$

The final answer is $\boxed{ (0, \frac{197}{4}) }$