  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 (GSMF),
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 (sSFRs), 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 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, $z\sim0.05$ are consistent with other
analyses. However, our inferred cosmic SFRDs peak at $z\sim0.4$, approximately
6 Gyr later than direct observations suggest, meaning 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 non-parametric SFH models.
