We propose a novel mode-locked fiber laser design which features a passive nonlinear triangular pulse formation and self-similar parabolic pulse amplification intra cavity. Attribute to the nonlinear reshaping progress in the passive fiber, a triangular-profiled pulse with negative-chirp is generated and paved the way for rapid and efficient self-similar parabolic evolution in a following short-length high-gain fiber. In the meanwhile, the accompanied significantly compressed narrow spectrum from this passive nonlinear reshaping also gives the promise of pulse stabilization and gain-shaping robustness without strong filtering. The resulting short average intra-cavity pulse duration, low amplified spontaneous emission (ASE) and low intra-cavity power loss are essential for the low-noise operation. Simulations predict this modelocked fiber laser allows for high-energy ultra-short transform-limited pulse generation exceeding the gain bandwidth. The output pulse has a de-chirped duration (full-width at half maximum, FWHM) of 27 fs. In addition to the ultrafast laser applications, the proposed fiber laser scheme can support low-noise parabolic and triangular pulse trains at the same time, which are also attractive in optical pulse shaping, all-optical signal processing and high-speed communication applications.