  We complete our boundedness theory of commutators of bilinear bi-parameter
singular integrals by establishing the following result. If $T$ is a bilinear
bi-parameter singular integral satisfying suitable $T1$ type assumptions,
$\|b\|_{\operatorname{bmo}(\mathbb{R}^{n+m})} = 1$ and $1 < p, q \le \infty$
and $1/2 < r < \infty$ satisfy $1/p+1/q = 1/r$, then we have $$ \|[b, T]_1(f_1,
f_2)\|_{L^r(\mathbb{R}^{n+m})} \lesssim \|f_1\|_{L^p(\mathbb{R}^{n+m})}
\|f_2\|_{L^q(\mathbb{R}^{n+m})}. $$ Previously the range $r \le 1$ was proved
only in the paraproduct free situation. The main novelty lies in the treatment
of the so called partial paraproducts.
