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Source. Published p. 283, Theorem 10.2 (PDF).

Statement. If every cross intersection of $\mathcal A,\mathcal B\subseteq2^{[n]}$ has parity $i\in\{0,1\}$, then $|\mathcal A||\mathcal B|\le2^n$ for $i=0$ and at most $2^{n-1}$ for $i=1$. In the even case, equality forces both sets of characteristic vectors to be full mutually orthogonal linear subspaces over $\mathbb F_2$.

Proof. Empty families are immediate. In the even case their linear spans $V,W\subseteq\mathbb F_2^n$ are orthogonal, so $\dim V+\dim W\le n$. Consequently $|\mathcal A||\mathcal B|\le|V||W|=2^{\dim V+\dim W}\le2^n$. Equality forces equality in both inclusions, giving the stated full subspaces. In particular both families contain the empty set.

In the odd case append a coordinate one to every characteristic vector on both sides. The new spans in $\mathbb F_2^{n+1}$ are orthogonal. The last-coordinate functional is nonzero on each span, so its level one contains exactly half the span. The two families lie in these two halves, yielding $|\mathcal A||\mathcal B|\le2^{\dim V+\dim W-2}\le2^{n-1}$. $\square$