Tarpey broke into the first team following loan spells with Walton & Hersham and Chertsey Town, before joining Farnborough in June 2012 after turning down a new contract at the Beveree.
[3][4] Tarpey scored 13 league goals in eighteen months at the Yellows before his contract was cancelled in January 2014 by mutual consent.
[8] He opened his first season in the National League with seven goals in his first six games, before signing for Barnet on 31 August 2017 which enabled him to leave his full-time job as a fire & security alarm installer.
Tarpey turned down the offer of a new contract from the Bees and left the club at the end of the 2018–19 season in order to return to part-time football.
[20] On 15 November 2023, Leatherhead announced that Tarpey was leaving the club amicably and by mutual consent, with the player's desire to get more first team football being a key reason.