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,