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We give a cohomological criterion for a parabolic vector bundle on a curve tobe semistable. It says that a parabolic vector bundle $E$ with rationalparabolic weights is semistable if and only if there is another parabolicvector bundle $F$ with rational parabolic weights such that the cohomologies ofthe vector bundle underlying the para- bolic tensor product $E \otimes F$vanish. This criterion generalizes the known semistability criterion ofFaltings for vector bundles on curves and significantly improves the result in[Bis07].

Semistability criterion for parabolic vector bundles on curves