Example: Correlated noise preserves parity perfectly

Keep $X$ uniform on ${\Omega }_2$, but let the two noise signs be one common sign:

Each coordinate separately still has crossover $1/4$, but the flips are not independent. Then

Consequently parity has mutual information one bit with $Y$, and its posterior mean has absolute value one. Its Hellinger loss is therefore one, not at most $1-\sqrt{3}/2$.