They arise in analytic theory of differential equations with regular singularities, Riemann Hilbert problem and Painleve equations.

MR1924757 
Biswas, Indranil
A criterion for the existence of a flat connection on a parabolic vector bundle. (English summary) 
Adv. Geom. 2 (2002), no. 3, 231–241.

MR1488348  Arinkin, D.; Lysenko, S. On the moduli of SL(2)-bundles with connections on P1∖{x1,⋯,x4}. Internat. Math. Res. Notices 1997, no. 19, 983–999.