In the sense that birds are dinosaurs, and thus that dinosaurs are still alive, yes. However they were not alive when non-avian dinosaurs were around. Parrots did not evolve until a few million years after non-avian dinosaurs died out.
Dinosaurs as most people think of them are completely extinct. However, birds evolved from dinosaurs, and as such, are considered to be a group of dinosaurs. Since birds are still alive today, so are some dinosaurs. No dinosaurs other than birds are alive today, though.