We present a deterministic $O(\log \log \log n)$-round low-space Massively Parallel Computation (MPC) algorithm for the classical problem of $(Δ+1)$-coloring on $n$-vertex graphs. In this model, every machine has a sublinear local memory of size $n^φ$ for any arbitrary constant $φ\in (0,1)$. Our algorithm works under the relaxed setting where each machine is allowed to perform exponential (in $n^φ$) local computation, while respecting the $n^φ$ space and bandwidth limitations. Our key technical contribution is a novel derandomization of the ingenious $(Δ+1)$-coloring LOCAL algorithm by Chang-Li-Pettie (STOC 2018, SIAM J. Comput. 2020). The Chang-Li-Pettie algorithm runs in $T_{local}=poly(\log\log n)$ rounds, which sets the state-of-the-art randomized round complexity for the problem in the local model. Our derandomization employs a combination of tools, most notably pseudorandom generators (PRG) and bounded-independence hash functions. The achieved round complexity of $O(\log\log\log n)$ rounds matches the bound of $\log(T_{local})$, which currently serves an upper bound barrier for all known randomized algorithms for locally-checkable problems in this model. Furthermore, no deterministic sublogarithmic low-space MPC algorithms for the $(Δ+1)$-coloring problem were previously known.