The Broadway show, which tells the story of a Mormon missionary group as they travel to Africa, won nine awards in total out of the 14 it was nominated for.
The Book of Mormon defeated Sister Act, Catch Me If You Can and The Scottsboro Boys in the 'Best Musical' category.
Parker accepted the award, saying: "Thank you. This has been really, really cool. We did this because we all secretly wanted to have a big happy Mormon family, and now we do... Andrew Rannells and Josh Gad (The Book of Mormon's principal actors), you guys are amazing.
"I know that on behalf of this big happy Mormon family, we want to thank the audiences because you made this show what you did. Therefore, you're going to have to atone for it."
The Book of Mormon was also recognised in a number of technical categories including 'Best Sound Design Of A Musical', 'Best Lighting Design Of A Musical' and 'Best Scenic Design Of A Musical'.
It also scooped the gongs for 'Best Orchestration', 'Best Original Score (Music And/Or Lyrics) Written For The Theatre', 'Best Direction Of A Musical' and 'Best Book Of A Musical'.
Nikki M. James, who plays Nabalungi, won the award for 'Best Performance By An Actress In A Featured Role In A Musical'.
A visibly emotional James thanked Parker, Stone and co-creator Robert Lopez for "writing the most incredible show, the funniest and most beautiful, heartfelt musical that's changing the face of American musical theatre".
She thanked several people including her fellow cast members, crew and close family before adding: "This gorgeous award is for my poppy, who isn't here with me tonight, but he sits on my shoulder every night."
Nick Stafford's adaptation of the Michael Morpurgo novel War Horse won 'Best Play' and claimed a further four awards.
Daniel Radcliffe's How to Succeed in Business Without Really Trying lost out to Anything Goes in the category 'Best Revival of a Musical'. However, Radcliffe's co-star John Laroquette took home the award for Best Performance By An Actor In A Featured Role In A Musical'.
Catherine Zeta-Jones, Whoopi Goldberg, Bono, The Edge, Brooke Shields and Hugh Jackman also made appearances during the telecast, which was hosted by How I Met Your Mother star Neil Patrick Harris.
Jackman and Harris also performed a comedic musical duet in which they sparred about their ability to successfully host awards shows.
Watch Neil Patrick Harris and Hugh Jackman's Tony Awards duet below: