There is a simple general argument showing that $\xi$ is trivial: Take any homogeneous space $G/H$ and any representation $V$ of $G$ and restrict the representation to $H$. Then the homogeneous vector bundle $G\times_H V\to G/P$ is a trivial as a vector bundle. A trivialization can be written down explicitly. It is induced by the map $G\times V\to (G/H)\times V$ defined by $(g,v)\mapsto (gH,g\cdot v)$. This is evidently $H$-invariant and factorizes to an isomorphism $G\times_H V\to (G/H)\times V$ of vector bundles.

There is a simple general argument showing that $\xi$ is trivial: Take any homogeneous space $G/H$ and any representation $V$ of $G$ and restrict the representation to $H$. Then the homogeneous vector bundle $G\times_H V\to G/P$ is a trivial as a vector bundle. A trivialization can be written down explicitly. It is induced by the map $G\times V\to (G/H)\times V$ defined by $(g,v)\mapsto (gH,g\cdot v)$. This is evidently $H$-invariant and factorizes to an isomorphism $G\times_H V\to (G/H)\times V$ of vector bundles.

Edit (following the comments by @Sebastian_Goette and @Asghar_Ghorbanpour): This argument can be applied both to the three dimensional representation of $A_4$ called $V$ in the question and to the representation on $\mathbb R^4$ by permutation of coordinates. The latter representation also extends to $SO(3)$ as the direct sum of the representation on $V$ and a trivial representation.