We establish 2-jet determinacy for the symmetry algebra of the underlying structure of any (complex or real) parabolic geometry. At non-flat points, we prove that the symmetry algebra is in fact 1-jet determined. Moreover, we prove 1-jet determinacy at any point for a variety of non-flat parabolic geometries - in particular torsion-free, parabolic contact, and several other classes.

Some historical remarks and references added; text further polished and minor corrections made. The main changes concern: some formulations in the introduction, Remark 3.9, Theorem 6.6, Section 7.3