In the 1980s he won one ranking tournament, but his peak came in the early 1990s. He dominated the season 1990-1991 with profit of two rankings and the World Championship 1991. Further in the 90 's, he has a few more rankings won, but since then he no longer runs at the top. Yet he still as qualifier tournaments take part; in 2005 he failed for the first time in the main event of the World Championships.
In 1996, Parrot appointed member in the order of the British Empire and should therefore since then the abbreviation MBE (Member of the Order of the British Empire ") run behind his name.
- World Snooker Championship (1991)
- European Open (1989, 1990, 1996)
- Dubai Duty Free Classic (1991, 1992)
- UK Championship (1991)
- International Open (1994)
- Thailand Classic (1995)