  We explicitly calculate the Hausdorff dimension of the graph and range of an
isotropic stable L\'{e}vy process $X$ plus deterministic drift function $f$.
For that purpose we use a restricted version of the genuine Hausdorff dimension
which is called the parabolic Hausdorff dimension. It turns out that covers by
parabolic cylinders are optimal for treating self-similar processes, since
their distinct non-linear scaling between time and space geometrically matches
the self-similarity of the processes. We provide explicit formulas for the
Hausdorff dimension of the graph and the range of $X+f$. In sum the parabolic
Hausdorff dimension of the drift term $f$ alone contributes to the Hausdorff
dimension of $X+f$. Further, we derive some formulas and bounds for the
parabolic Hausdorff dimension.
