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 estimate the cost of the optimal solution to a given routing problem. Obtaining accelerated $\varepsilon$ dependencies for both problems requires developing a stronger multicommodity cost approximator, one where cancellations between different commodities are disallowed. For maximum flow, we observe that a recent linear cost approximator due to Agarwal et al. (SODA 2024) can be augmented with additional parallel operations and achieve $\varepsilon^{-1}$ dependency via the box-simplex game. For transshipment, we also construct a deterministic and distributed approximator. This yields a deterministic CONGEST algorithm that requires $\tilde{O}(\varepsilon^{-1}(D + \sqrt{n}))$ rounds on general networks of hop diameter $D$ and $\tilde{O}(\varepsilon^{-1}D)$ rounds on minor-free networks to compute a $(1+\varepsilon)$-approximation.