Is a budgie a parrot

Yes, budgies are small parrots. Parrots are recognisable by their curved beaks and sharp claws, which are often used to hold foods or other items of interest to the bird.

Away from their native Australia, budgerigars are sometimes called "parakeets", but this term applies to a variety of long-tailed small to medium sized members of the parrot family, and does not exclusively mean budgies.