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This paper reports a comprehensive study of pump-to-signal intensity modulation transfer (IMT) in single-pump fiber optic parametric amplifiers (FOPAs). In particular, the IMT is studied for the first time for high-frequency fluctuations of the pump as well as in the saturated gain regime. The IMT cut-off frequency in typical single-pump FOPAs is around 100-200 GHz. The possibilities to shift this frequency based on dispersion and nonlinearities involved in the parametric gain are discussed. The severe IMT to the signal at low modulation frequencies can be suppressed by more than 50% in the gain saturation regime with respect to the linear gain operation. Experimental results confirm the validity of the numerical study.