In trying to answer my question here, I think it suffices to show that $T(G/P)$ has fibre $\mathfrak{g}/\mathfrak{p}$ over each point $gP \in G/P$, i.e. the title. Here, $P \subset G$ is any Lie subgroup, and the frakturs their corresponding Lie algebras.
My outline of ideas is that since $TG \cong G \times T_eG$, then $T(G/P) \cong G/P \times T_e(G/P) = G/P \times \mathfrak{g}/\mathfrak{p}$, so that $T(G/P)$ has fibre $\mathfrak{g}/\mathfrak{p}$ over each point $gP$.
However I think this is too simple, and am not sure if I am missing something else.