We study the Pareto Set Identification (PSI) problem in a structured multi-output linear bandit model. In this setting, each arm is associated a feature vector belonging to $\mathbb{R}^h$, and its mean vector in $\mathbb{R}^d$ linearly depends on this feature vector through a common unknown matrix $Θ\in \mathbb{R}^{h \times d}$. The goal is to identify the set of non-dominated arms by adaptively collecting samples from the arms. We introduce and analyze the first optimal design-based algorithms for PSI, providing nearly optimal guarantees in both the fixed-budget and the fixed-confidence settings. Notably, we show that the difficulty of these tasks mainly depends on the sub-optimality gaps of $h$ arms only. Our theoretical results are supported by an extensive benchmark on synthetic and real-world datasets.