  Answering a key point left open in a recent work of Bongers, Guo, Li and
Wick, we provide the following lower bound
  $$
  \|b\|_{\text{BMO}_{\gamma}(\mathbb{R}^2)}\lesssim
\|[b,H_{\gamma}]\|_{L^p(\mathbb{R}^2)\to L^p(\mathbb{R}^2)},
  $$ where $H_{\gamma}$ is the parabolic Hilbert transform.
