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The equation is parabolic at a point when $\boxed{b^2-ac=0}$ and at least one of $a,b,c$ is nonzero. The principal matrix then has rank one, with a repeated characteristic direction. If all three vanish, there is no second-order principal part there and the usual parabolic classification does not apply.

Take pairwise disjoint coordinate discs $U_m$ around $x_m$ and put $U_0=S\setminus \{x_1,\dots ,x_N\}$. A prescribed principal part of a meromorphic function $P_m$ is holomorphic on $U_0\cap U_m$. Definewith all other components zero. No three distinct cover members meet, so the Čech cocycle condition is automatic. Thus $(P_{ij})$ is a Čech one-cocycle for $\mathcal{O}$.