Parametric models for galaxy star formation histories (SFHs) are widely used, though they are known to impose strong priors on physical parameters. This has consequences for measurements of the galaxy stellar-mass function, star formation rate density (SFRD), and star-forming main sequence (SFMS). We investigate the effects of the exponentially declining, delayed exponentially declining, lognormal, and double power-law SFH models using BAGPIPES. We demonstrate that each of these models imposes strong priors on specific star formation rates (SFRs), potentially biasing the SFMS, and also imposes a strong prior preference for young stellar populations. We show that stellar mass, SFR, and mass-weighted age inferences from high-quality mock photometry vary with the choice of SFH model by at least 0.1, 0.3, and 0.2 dex, respectively. However, the biases with respect to the true values depend more on the true SFH shape than the choice of model. We also demonstrate that photometric data cannot discriminate between SFH models, meaning that it is important to perform independent tests to find well-motivated priors. We finally fit a low-redshift, volume-complete sample of galaxies from the Galaxy and Mass Assembly (GAMA) Survey with each model. We demonstrate that our stellar masses and SFRs at redshift z ~ 0.05 are consistent with other analyses. However, our inferred cosmic SFRDs peak at z ~ 0.4, approximately 6 Gyr later than direct observations suggest, meaning that our mass-weighted ages are significantly underestimated. This makes the use of parametric SFH models for understanding mass assembly in galaxies challenging. In a companion paper, we consider nonparametric SFH models.