STO-3G has exponent parameters which can be found in a reference book. Can these parameters be calculated using, for instance, a Hartree–Fock equation for one atom? Or using another way?
You are right that the STO-nG basis sets are optimized separately for each individual element (as an isolated atom, rather than as an ion or an atom within a bigger molecule). The exponents are obtained by by a least squares fit of the Gaussian orbitals to a single Slater-type orbital (STO).
For other basis sets (not named STO-nG but named something else, like cc-pCVDZ), the exponents are calculated by doing Hartree-Fock and CISD calculations, for example.
If you want to know more about how Gaussian basis sets are optimized, you might find the following thread interesting: Optimization of Gaussian basis sets within the Hartree-Fock Method.
There's also been some discussion about STO-3G specifically and why different sources have different exponent values for STO-3G: Difference between original STO-3G and BSE STO-3G