We report on a simple passive scheme to reduce the intensity noise of high-power nonlinear fiber amplifiers by use of the spectral-breathing parabolic evolution of the pulse amplification with an optimized negative initial chirp. In this way, the influences of amplified spontaneous emission (ASE) on the amplifier intensity noise can be efficiently suppressed, owing to the lower overall pulse chirp, shorter spectral broadening distance, as well as the asymptotic attractive nature of self-similar pulse amplification. Systematic characterizations of the relative intensity noise (RIN) of a free-running nonlinear Yb-doped fiber amplifier are performed over a series of initial pulse parameters. Experiments show that the measured amplifier RIN increases respect to the decreased input pulse energy, due to the increased amount of ASE noise. For pulse amplification with a proper negative initial chirp, the increase of RIN is found to be smaller than with a positive initial chirp, confirming the ASE noise tolerance of the proposed spectral-breathing parabolic amplification scheme. At the maximum output average power of 27W (25-dB amplification gain), the incorporation of an optimum negative initial chirp (-0.84 chirp parameter) leads to a considerable amplifier root-mean-square (rms) RIN reduction of ~20.5% (integrated from 10 Hz to 10 MHz Fourier frequency). The minimum amplifier rms RIN of 0.025% (integrated from 1 kHz to 5 MHz Fourier frequency) is obtained along with the transform-limited compressed pulse duration of 55fs. To our knowledge, the demonstrated intensity noise performance is the lowest RIN level measured from highpower free-running femtosecond fiber amplifiers.