AMC 12 2016b Problem 6

All three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$?

\textbf{(A)}\ 4 \textbf{(B)}\ 6 \textbf{(C)}\ 8 \textbf{(D)}\ 10 \textbf{(E)}\ 16