This work considers the problem of detecting unknown inputs in networked systems whose dynamics are governed by time-varying unknown parameters. We propose a strategy in opposition to the commonly employed approach of first estimating the unknown parameters and then using the estimates as the true parameter values for detection, e.g. maximum-likelihood approaches. The suggested detection scheme employs test statistics that are invariant to the unknown parameters and do not rely on parameter estimation. We specifically consider the case of severe lack of prior knowledge, i.e., the problem of detecting unknown inputs when nothing is known of the system but some primitive structural properties, namely that the system is a linear network, subject to Gaussian noise, and that a certain input signal is either present or not. The aim is thus to analyze the structure and performances of invariant tests in a limiting case, specifically where the amount of prior information is minimal. The developed test is proven to be maximally invariant to the unknown parameters and Uniformly Most Powerful Invariant (UMPI). Simulation results indicate that for arbitrary networked systems the parameterinvariant detector achieves a specified probability of false alarm while ensuring that the probability of detection is maximized.