Let $A$, $B$, and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle (right angle at $C$) with area $100$. What is the sum of the digits of the $y$-coordinate of $C$?

1. Writing $A=(-a,a^2), B=(a,a^2), C=(c,c^2)$ with the right angle at $C$, the perpendicularity condition forces $a^2-c^2=1$ regardless of the area.
2. The area is $\frac12(2a)(a^2-c^2)=a$, so $a=100$ and $c^2=a^2-1=9999$.
3. Digit sum of $9999$ is $9+9+9+9=\boxed{36}$.