The equation of the parabola given is \( y = x^2 \).
This is a standard quadratic function, where the coefficient of \( x^2 \) is positive (specifically, 1). Since the parabola opens upwards, its minimum value occurs at the vertex.
The vertex of the parabola given by \( y = x^2 \) can be found at \( x = 0 \).
To find the minimum value of \( y \):
\[ y = (0)^2 = 0 \]
Therefore, the minimum value of the parabola \( y = x^2 \) is \( \boxed{0} \).