Coupled oscillators are one of the basic models in nonlinear dynamics. Here, we study numerically and analytically the spectra of two harmonic oscillators with stochastically fluctuating coupling and driving forces reproducing a thermostat. We show that, even at small coupling, vanishing on average, the oscillation spectra exhibit mixing, even though no cross-correlations exists between the oscillators. Our results reveal a new mechanism of mode mixing for stochastically uncorrelated systems that is crucial for analysis of spectra in various systems, from simple liquids to living systems.