  We develop, validate and apply a forward model to estimate stellar
atmospheric parameters ($T_{\rm eff}$, $\log{g}$ and $\mathrm{[Fe/H]}$),
revised distances and extinctions for 220 million stars with XP spectra from
$\textit{Gaia}$ DR3. Instead of using $\textit{ab initio}$ stellar models, we
develop a data-driven model of $\textit{Gaia}$ XP spectra as a function of the
stellar parameters, with a few straightforward built-in physical assumptions.
We train our model on stellar atmospheric parameters from the LAMOST survey,
which provides broad coverage of different spectral types. We model the
$\textit{Gaia}$ XP spectra with all of their covariances, augmented by 2MASS
and WISE photometry that greatly reduces degeneracies between stellar
parameters, yielding more precise determinations of temperature and dust
reddening. Taken together, our approach overcomes a number of important
limitations that the astrophysical parameters released in $\textit{Gaia}$ DR3
faced, and exploits the full information content of the data. We provide the
resulting catalog of stellar atmospheric parameters, revised parallaxes and
extinction estimates, with all their uncertainties. The modeling procedure also
produces an estimate of the optical extinction curve at the spectral resolution
of the XP spectra ($R \sim 20-100$), which agrees reasonably well with the
${R(V) = 3.1}$ CCM model. Remaining limitations that will be addressed in
future work are that the model assumes a universal extinction law, ignores
binary stars and does not cover all parts of the Hertzsprung-Russell Diagram
($\textit{e.g.}$, white dwarfs).
