The father of archaeology is William Flinders Petrie who is most legitimately known as the official Father of Archaeology. The science dates way back though. There has been a myriad of people who have contributed to the discipline.
The 5th century-BCE famous Greek historian Herodotus actually was the first ever scholar to place the practice into a system and make it into the science that we know it today. But, for all intents and purposes, Petrie stands as the one whom most give the tribute of being the first and utmost forefather in the field.