Measurements of the top quark-antiquark pair production cross section, $\sigma_\mathrm{t\bar{t}}$, can be used to constrain the strong coupling constant, $\alpha_\mathrm{S}$, the top quark mass, $m_\mathrm{t}$, and the parton distribution functions (PDFs). In this poster, the two most recent relevant results published by the CMS Collaboration are presented. The analyses are performed using proton-proton collision data at a centre-of-mass energy of 13 TeV recorded by the CMS detector at the CERN LHC in 2016, corresponding to an integrated luminosity of $35.9~\mathrm{fb^{-1}}$. In the first one, $\alpha_\mathrm{S}$ and $m_\mathrm{t}$ are extracted independently from a measurement of the inclusive $\sigma_\mathrm{t\bar{t}}$, using next-to-next-to-leading order theoretical predictions. In the second, a measurement of the normalized triple-differential $\mathrm{t\bar{t}}$ cross section is performed; the result is then used together with HERA deep-inelastic scattering data to perform a simultaneous determination of $\alpha_\mathrm{S}$, $m_\mathrm{t}$, and the PDFs, at next-to-leading order. As a result, the uncertainty in the gluon PDF and its correlation with $\alpha_\mathrm{S}$ are significantly reduced at high parton momentum fraction, the kinematic range probed by $\mathrm{t\bar{t}}$ production. The result also yields the most precise determination of the top quark pole mass, to date.