We show that a quadratic system of pseudofermions, with tunable fractionalized statistics, can host a rich phase diagram on a one-dimensional chain with nearest- and next-nearest-neighbor hopping. Using a combination of numerical and analytical techniques, we show that by varying the statistical angle and the ratio of the hopping, the system stabilizes two Tomonaga-Luttinger liquids (TLL) with central charges c = 1 and 2, respectively, along with the inversion symmetry broken bond-ordered (BO) insulating phase. Interestingly, the two quantum phase transitions in the system, (1) between the two TLLs and (2) the c = 1 TLL and BO phase, can be engendered by solely tuning the statistics of the pseudofermions. Our analysis shows that both these transitions are continuous and novel with the former lacking a local order-parameter based description and the latter of Berezinskii-Kosterlitz-Thouless type. These phases and phase transitions can be of direct experimental relevance in the context of recent studies of fermionic cold atoms.