The definition on OALD is identical for parcel and packet.
parcel (especially British English) (North American English usually package) something that is wrapped in paper or put into a thick envelope so that it can be sent by mail, carried easily, or given as a present
packet a small object wrapped in paper or put into a thick envelope so that it can be sent by mail, carried easily or given as a present
According to OALD, package is a North American term, though ngram viewer indicates that it is also the most common word in British English.
What is the difference between packet, parcel and package? Do the words denote size or material (package in its second meaning)?