This presentation introduces an optimization-based methodology that yields, within a unified framework, both a parallel operator splitting technique and a domain decomposition method. The approach is designed for coupled systems of equations posed on mixed-dimensional domains. These problems are characterized by the presence of multiple PDEs defined on domains formed by the union of geometrical entities of different spatial dimensions (3D–2D–1D coupled domains). The proposed methodology addresses both the large size of the resulting system of equations, through a decoupling of the governing equations, and the geometrical complexity of the domain, by means of a domain decomposition strategy. Unlike many existing operator splitting approaches, the decoupled equations can be solved in parallel, and domain decomposition techniques can be applied independently to each of them. First, suitable internal variables are introduced to decouple the equations in the system. These variables act as control variables in an optimization problem, which leads to the parallel operator splitting formulation. Subsequently, the optimization strategy is applied again to each decoupled equation in order to perform domain decomposition. In this case, additional internal variables are introduced at the geometrical interfaces of the domain, and the minimization of an appropriate functional enforces the matching conditions. The main advantages of the proposed method lie in the ability to choose the discretization of the additional variables independently of the other unknowns, and in the inherently parallel structure of both the operator splitting and the domain decomposition procedures. Application examples are proposed for the Biot equation for port-mechanics and for the growth of capillary vessels in tissues.