Classic symmetry-breaking problems on graphs have gained a lot of attention in models of modern parallel computation. The Adaptive Massively Parallel Computation (AMPC) is a model that captures the central challenges in data center computations. Chang et al. [PODC'2019] gave an extremely fast, constant time, algorithm for the $(Δ+ 1)$-coloring problem, where $Δ$ is the maximum degree of an input graph of $n$ nodes. The algorithm works in the most restrictive low-space setting, where each machine has $n^δ$ local space for a constant $0 < δ< 1$. In this work, we study the vertex-coloring problem in sparse graphs parameterized by their arboricity $α$, a standard measure for sparsity. We give deterministic algorithms that in constant, or almost constant, time give $\text{poly} ~α$ and $O(α)$-colorings, where $α$ can be arbitrarily smaller than $Δ$. A strong and standard approach to compute arboricity-dependent colorings is through the Nash-Williams forest decomposition, which gives rise to an (acyclic) orientation of the edges such that each node has a small out-degree. Our main technical contribution is giving efficient deterministic algorithms to compute these orientations and showing how to leverage them to find colorings in low-space AMPC. A key technical challenge is that the color of a node may depend on almost all of the other nodes in the graph and these dependencies cannot be stored on a single machine. Nevertheless, our novel and careful exploration technique yields the orientation, and the arboricity-dependent coloring, with a sublinear number of adaptive queries per node.