  In this note, we establish a novel maximal inequality of the 2D Young
integral $\int_a^b\int_c^d FdG$ in terms of the $(p,q)$-bivariation norms of
the section functions $x\mapsto F(x,y)$ and $y\mapsto F(x,y)$ where
$G:[a,b]\times [c,d]\rightarrow \mathbb{R}$ is a controlled path satisfying
finite $(p,q)$-variation conditions. The proof is reminiscent from the Young's
original ideas \cite{young1} in defining two-parameter integrals in terms of
$(p,q)$-finite bivariations. Our result complements the standard maximal
inequality established by Towghi \cite{towghi1} in terms of joint variations.
We apply the maximal inequality to get novel strong approximations for 2D Young
integrals w.r.t the Brownian local time in terms of number of upcrossings of a
given approximating random walk.
