We combine several recent advancements to solve $(1+\varepsilon)$-transshipment and $(1+\varepsilon)$-maximum flow with a parallel algorithm with $\tilde{O}(1/\varepsilon)$ depth and $\tilde{O}(m/\varepsilon)$ work. We achieve this by developing and deploying suitable parallel linear cost approximators in conjunction with an accelerated continuous optimization framework known as the box-simplex game by Jambulapati et al. (ICALP 2022). A linear cost approximator is a linear operator that allows us to efficiently es…