The answer is $k=14$. For a valid construction, place the black rooks on cells $(a, a)$ for $2 \leq a \leq 7$ and the white rooks on cells $(a, a+1)$ and $(a+1, a)$ for $1 \leq a \leq 7$. Now, we prove the optimality. As rooks can only attack opposite color rooks, the color of rooks in each row is alternating. The difference between the number of black and white rooks is thus at most the number of rooks. Thus, $k \leq 6+8=14$.