Running until 9 November as part of Lloyd's Trafalgar Transformed season, the play, which stars Hayley Atwell, Harry Hadden-Paton, Al Weaver and Mathew Horne, contrasts attitudes to homosexuality in the 1950s and the present day.
Access to the event is free for ticket holders to that evening's performance, with £10 of every ticket sold going to Stonewall.
"We know that too many children still hear homophobic abuse every single day," Young said. "The Pride is an inspirational tale of how far the gay community has come but we know that there's still so much to be done.
"That's why it's fantastic that the crew and cast are raising money to support Stonewall's work in schools across Britain and I'm delighted to be part of this important debate on homophobic language."
Other events include a masterclass with the play's author Alexi Kaye Campbell (on 21 October), and a series of 'late night Fridays' on 4, 11 and 18 October featuring the London Gay Men's Chorus, 4 Poofs and a Piano and stand-up Scott Capurro.
There will also be a staged reading of Wildeblood (3 October), a new play by Jonathan Cooper based on transcripts from the 1954 trial of diplomatic correspondent Peter Wildeblood who was jailed for 18 months for taking part in homosexual activity.
Jamie Lloyd said: "As a part of our commitment to engage directly with the world outside our theatre's walls, we are thrilled to present a mini-festival of diverse events that examine, discuss and celebrate contemporary gay life and LGBT politics. I am excited to open up our season space to the world of comedy and cabaret for the very first time too."