We identify a new class of decidable hybrid automata: namely, parallel compositions of semi-algebraic o-minimal automata. The class we consider is fundamental to hierarchical modeling in many exemplar systems, both natural and engineered. Unfortunately, parallel composition, which is an atomic operator in such constructions, does not preserve the decidability of reachability. Luckily, this paper is able to show that when one focuses on the composition of semi-algebraic o-minimal automata, it is possible to translate the decidability problem into a satisfiability problem over formulæinvolving both real and integer variables. While in the general case such formulæ would be undecidable, the particular format of the formulæ obtained in our translation allows combining decidability results stemming from both algebraic number theory and first-order logic over (ℝ, 0, 1, +,*, <) to yield a novel decidability algorithm. From a more general perspective, this paper exposes many new open questions about decidable combinations of real/integer logics.