Parrett is without a club after being released by Spurs this summer.
The 21-year-old midfielder, who has recently had trials at Notts County and Plymouth, is set to play for the Rams' development squad against Crewe on Monday afternoon.
Parrett made his first-team debut for Tottenham in a UEFA Cup match against Shakhtar Donetsk in February 2009 and signed a three-year contract the following year.
He made three further appearances for the Premier League club, all of which came in the UEFA Cup or the Europa League.
Parrett has had loan spells at Aldershot, Plymouth, Charlton, Yeovil and Swindon, making 36 appearances in total.