In mathematics, the world parabolic geometry is used in the following meanings:

Euclidean geometry

More generally, a manifold with a connection having zero scalar curvature (the word elliptic would indicate positive scalar curvature and the word hyperbolic indicates negative scalar curvature)

Cartan geometry of parabolic type. This is a Cartan geometry modeled on the pair (G,P) where G is a semisimple Lie group and P a parabolic subgroup. Surprisingly, the above mentioned Euclidean geometry is not among these geometries, so this may cause confusion.