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Reference governors are add-on schemes that are used to modify trajectories to prevent controlled dynamical systems from violating constraints and so are playing an increasingly important role in aerospace, robotic, and other engineering applications. Here we present a novel reference governor design for systems whose polynomial constraints depend on unknown bounded parameters. This is a significant departure from earlier treatments of reference governors, where the constraints were linear or known, because here we transfer the uncertainties into the constraints instead of having them in the closed loop dynamics, which greatly simplifies the task of determining future evolution of the constraints. Unlike our earlier treatment of reference governors with polynomial constraints, which transformed the constraints into linear ones that depend on an augmented state of the system, here we transform the constraints into linear ones that depend on both the system's state and uncertain parameters. Convexity allows us to compute the maximal output admissible set for an uncertain pre-stabilized linear system. We show that it is sufficient to only consider the extreme values of the uncertain parameters when computing and propagating the polynomial constraints. We illustrate our method using an uncertain longitudinal dynamics for civilian aircraft, which is controlled using a disturbance compensation method and needs to satisfy input and state constraints, and where our reference governor method ensures that safety constraints are always satisfied.