When a parabola is scaled vertically by a factor of \(\frac{1}{10}\), it means that the \(y\)-values of the function are multiplied by \(\frac{1}{10}\).
The original equation of the parabola is:
\[ y = x^2 \]
To apply the vertical scaling, we multiply the right side of the equation by \(\frac{1}{10}\):
\[ y = \frac{1}{10} x^2 \]
Thus, the equation of the new parabola after the vertical scaling is:
\[ \boxed{y = \frac{1}{10} x^2} \]