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The paper introduces two methodological extensions of a design-space dimensionality reduction method, namely the parametric model embedding (PME), that is particularly suitable for integration with CAD/CAE parametric models and was developed for shape optimization in earlier work. The present developments extend the use of PME to structural optimization problems, paving the way for efficient design-space dimensionality reduction in complex fluid-structure interaction problems. PME is further extended with the aim of making the methodology more efficient in representing design variations that are potentially optimal (goal-oriented PME, GO-PME). PME and GO-PME are demonstrated for the design-space dimensionality reduction and multi-objective optimization of a structural unit problem, namely a simply supported beam under uniform load.