In this work, we study the class of problems solvable by (deterministic) Adaptive Massively Parallel Computations in constant rounds from a computational complexity theory perspective. A language L is in the class AMPC⁰ if, for every ε > 0, there is a deterministic AMPC algorithm running in constant rounds with a polynomial number of processors, where the local memory of each machine s = O(N^ε). We prove that the space-bounded complexity class ReachUL is a proper subclass of AMPC⁰. The complexity class ReachUL lies between the well-known space-bounded complexity classes Deterministic Logspace (DLOG) and Nondeterministic Logspace (NLOG). In contrast, we establish that it is unlikely that PSPACE admits AMPC algorithms, even with polynomially many rounds. We also establish that showing PSPACE is a subclass of nonuniform-AMPC with polynomially many rounds leads to a significant separation result in complexity theory, namely PSPACE is a proper subclass of EXP^{Σ₂^{𝖯}}.