To elaborate on my comment: yes, there is an Oka-Grauert principle for homogeneous spaces. The definite reference (besides the earlier papers of Grauert and Gromov) is the book "Stein manifolds and holomorphic mappings" by Forstneric.
In there, you find Corollary 5.4.8 telling you that for a holomorphic fiber bundle  $\pi:Z\to X$ over a reduced Stein space $X$ with Oka manifold fibers, the inclusion of holomorphic into continuous sections is a weak equivalence.  Proposition 5.1.1 tells you that every complex homogeneous manifold is an Oka manifold (via the exponential spray from the corresponding complex Lie group). Combining these two, any continuous section of a holomorphic fiber bundle with homogeneous space fibers can be deformed to a holomorphic section. Therefore, a holomorphic fiber bundle over the punctured disc with homogeneous space fibers is holomorphically trivial if it is topologically trivial. However, as pointed out by abx, not every holomorphic fiber bundle over the punctured disc is topologically trivial.