  Let $X$ be a quasi-projective curve, compactified to $(Y,D)$ with $X=Y-D$. We
construct a Deligne-Hitchin twistor space out of moduli spaces of framed
$\lambda$-connections of rank $2$ over $Y$ with logarithmic singularities and
quasi-parabolic structure along $D$. To do this, one should divide by a
Hecke-gauge groupoid. Tame harmonic bundles on $X$ give preferred sections, and
the relative tangent bundle along a preferred section has a mixed twistor
structure with weights $0,1,2$. The weight $2$ piece corresponds to the
deformations of the KMS structure including parabolic weights and the residues
of the $\lambda$-connection.
