Building digital twins for parametric systems can become unaffordable in large-scale, high-dimensional settings, where high-quality snapshots are expensive to generate. Common remedies include data augmentation techniques to enrich sparsely populated datasets [1], local surrogates based on domain decomposition to reduce coupled degrees of freedom in space [2], and multi-fidelity strategies that mix simulations of different accuracy and cost [3]. In this context, multi-index stochastic collocation (MISC) represents an attractive solution because it combines many inexpensive low-fidelity runs with few costly high-fidelity ones. However, the accuracy of the resulting surrogate can degrade in the presence of noise in the simulation data. This issue is particularly critical when low-fidelity solutions use under-resolved meshes, larger time steps, or looser solver tolerances. Standard MISC then interpolates numerical errors, corrupts the response surface, and may introduce spurious high-frequency oscillations. In this talk, we present PlateauMISC, a noise-robust variant of MISC for parametric partial differential equations (PDEs) [4]. PlateauMISC monitors the coefficients of the polynomial expansion and it links solver noise to a plateau in their decay. The method detects when added fidelities are dominated by noise and stops refining in directions that no longer contribute meaningful information. This prevents overfitting of numerical artifacts while retaining the efficiency benefits of multi-fidelity modelling. Numerical experiments of parabolic advection-diffusion PDEs with uncertain coefficients and parametric turbulent incompressible Navier-Stokes problems will be presented. The results show improved accuracy and robustness with respect to standard MISC. They also show that PlateauMISC can extract useful information from under-resolved meshes that are otherwise unreliable in single-fidelity settings. [1] https://doi.org/10.1002/nme.7624 [2] https://doi.org/10.1016/j.cma.2023.116484 [3] https://doi.org/10.1007/s00366-021-01588-0 [4] https://doi.org/10.48550/arXiv.2507.03691