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

The twistor geometry of parabolic structures in rank two