  Given a holomorphic principal bundle $Q\, \longrightarrow\, X$, the universal
space of holomorphic connections is a torsor $C_1(Q)$ for $\text{ad} Q \otimes
T^*X$ such that the pullback of $Q$ to $C_1(Q)$ has a tautological holomorphic
connection. When $X\,=\, G/P$, where $P$ is a parabolic subgroup of a complex
simple group $G$, and $Q$ is the frame bundle of an ample line bundle, we show
that $C_1(Q)$ may be identified with $G/L$, where $L\, \subset\, P$ is a Levi
factor. We use this identification to construct the twistor space associated to
a natural hyper-K\"ahler metric on $T^*(G/P)$, recovering Biquard's description
of this twistor space, but employing only finite-dimensional, Lie-theoretic
means.
