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Weak gravitational lensing, or weak lensing, is one of the most powerful probes for dark matter and dark energy science, although it faces increasing challenges in controlling systematic uncertainties as \edit{the statistical errors become smaller}. The Point Spread Function (PSF) needs to be precisely modeled to avoid systematic error on the weak lensing measurements. The weak lensing biases induced by errors in the PSF model second moments, i.e., its size and shape, are well-studied. However, Zhang et al. (2021) showed that errors in the higher moments of the PSF may also be a significant source of systematics for upcoming weak lensing surveys. Therefore, the goal of this work is to comprehensively investigate the modeling quality of PSF moments from the $3^{\text{rd}}$ to $6^{\text{th}}$ order, and estimate their impact on cosmological parameter inference. We propagate the \textsc{PSFEx} higher moments modeling error in the HSC survey dataset to the weak lensing \edit{shear-shear correlation functions} and their cosmological analyses. We find that the overall multiplicative shear bias associated with errors in PSF higher moments can cause a $\sim 0.1 σ$ shift on the cosmological parameters for LSST Y10. PSF higher moment errors also cause additive biases in the weak lensing shear, which, if not accounted for in the cosmological parameter analysis, can induce cosmological parameter biases comparable to their $1σ$ uncertainties for LSST Y10. We compare the \textsc{PSFEx} model with PSF in Full FOV (\textsc{Piff}), and find similar performance in modeling the PSF higher moments. We conclude that PSF higher moment errors of the future PSF models should be reduced from those in current methods to avoid a need to explicitly model these effects in the weak lensing analysis.