Parker takes charge after Vincent Kompany left the relegated Clarets for Bayern Munich.
He has successfully led Fulham and Bournemouth into the Premier League in past stints. His last job was in 2023 with Club Brugge.
"I’m really pleased to be here," Parker, who has penned a three-year contract, said.
"I have been speaking with Burnley for some time now which has been a positive thing as I’ve now got a real feel for the people around the club.
“To be around the training ground now you start to get that feeling again and I can’t wait to get on the training field and start work.
“For us to be successful this year is the most important thing. We have to win and build a team the fans and club can be proud of. This team can represent every single one of them in that aspect and that’s the aim.”
Burnley chairman Alan Pace also said: "His vision for the future aligns with our goals and ambitions of returning to the Premier League as soon as possible.
“His track record of success along with a commitment to developing young talent makes him an ideal fit.”
“We are confident he can lead us to new heights and continue to build on the solid foundations laid at Turf Moor.”