What date did the Beatles first perform?
Between 17 August - 3 January they played in Indra Club in Hamburg. I think it was their first performance on 17th August 1960 with their name The Beatles.
I recently read that they appeared on Parr's show BEFORE they did on Ed Sullivan. What date did the Beatles appear on Jack Parr?
Probably the first performance by The Quarrymen (the earliest band formed by Lennon) was in July 1957 at a garden fete in Woolton, Liverpool. The Beatles became a "serious" band in 1960 when they obtained a booking to play in Hamburg at the Indra club; their first professional performances could be argued to be there.
Their manager Brian Epstein, he was the owner of a record shop near the Cavern in Liverpool where the Beatles would perform at every lunch hour and he heard of them by the people coming into the store and checked them out and they agreed for him to be their manager because he wanted to make them big