The normal-mode splitting in the optomechanical systems has attracted widely attention. This behavior of mode splitting arises due to the strong interaction between the oscillator and optical light circulating inside the cavity, when the system is in the resolved side band regime. Recently, many methods are proposed to produce the larger splitting by increasing the optomechanical coupling. It is proved that the observation of the normal-mode splitting is more accessible by placing the optical parametric amplifier inside the cavity, which is due mainly to the increase of the coupling between the oscillator and the cavity field. On the other hand, quadratic coupling can also increase the effective optomechanical coupling. Our paper is based on the above general background, and we study the important influence of quadratic optomechanical coupling and the parametric amplifier on normal-mode splitting in the optomechanical system with both linear and quadratic dispersion coupling. To investigate the normal-mode splitting effect, we calculate the output spectrum of the cavity and the spectrum of the mechanical oscillator displacement. Then, the calculated spectrum is plotted as a function of normalized frequency. The output spectrum of the cavity and the mechanical oscillator’s position spectrum all exhibit the double-peak structure. Thus, we can obtain the conclusions that both the cavity field and the mechanical field have normal-mode splitting behaviors. Further analysis of width of the normal-mode splitting peaks is analysed by discussing the spectra with different quadratic couplings and the parametric gain values. We find that the width of double-peak structure of the spectrum is increasing with the quadratic coupling, as well as the parametric gain. The physical explanation leading to such a phenomenon is that the increasing of quadratic coupling and the parametric gain aids in producing more number of cavity photons. As a result, this also leads to the increasing in effective optomechanical coupling. Quadratic coupling and the parametric gain have similar effects on the normal-mode splitting. In other words, the the normal-mode splitting behavior can be controlled by quadratic coupling or the parametric gain. Thus, It also proves the conclusion of the reference [Journal of Modern Optics 66(5): 494-501 (2019)]that the optomechanical systems with linear coupling and quadratic coupling can be seen as a alternative platforms to hybrid-optomechanical systems with the parametric amplifier.