Parker's successes in this division have come as a result of having an absolutely disgusting squad for this level.
At Fulham, he had a squad featuring Aleksander Mitrovic, Ivan Cavaleiro, Bobby Reid, Tom Cairney, Anthony Knockaert, and somehow only managed to get promoted through the playoffs. He then bolstered that squad for the PL with the likes of Ademola Lookman and Ruben Loftus-Cheek, and finished 18th and 11 points from safety. Sacked at the end of that season, replaced by Marco Silva, who proved the real strength of that squad the next season in the Championship - league champions, 90 points, 106 goals scored. They've obviously gone on and become an established PL side under him.
Joined Bournemouth for 21/22 - dragged them over the line for automatic promotion, with a squad featuring Lloyd Kelly (now at Juventus), Jefferson Lerma, Philip Billing, Ryan Christie and Lewis Cook among others. Made a few signings in the summer, including Marcos Senesi and Marcus Tavernier, threw a tantrum in front of the media about not being able to sign enough players after getting slapped 9-0 at Liverpool, and was gone before the end of August. Brought in Gary O'Neil, who led them to safety and a 15th placed finish, replaced him with Andoni Iraola in summer 2023, and never looked back.
And at Burnley, he finished 2nd on goal difference, despite only conceding 16 goals in 46 games (they scored 69 goals that season; for context, Norwich City finished 13th and scored more than that. And when Burnley won the league under Kompany in 22/23, they scored 87 goals). That team was largely reliant on James Trafford, Maxime Estevé and CJ Egan-Riley as an outrageous defensive base. And I'll concede here that Parker was extremely hard done by in that Egan-Riley didn't sign a new deal and left for Marseille on a free, and Trafford went back to Man City. No way to replace two-thirds of the best defence in Europe as they were that season. So it's of no real surprise that they've struggled this season, and are already relegated with multiple games to spare. A significantly weakened spine, and a manager with a very poor track record in the PL.
Considering that our squad isn't anywhere the strength of the 3 teams Parker has managed in the Championship, and he won't have significant funds to spend, I doubt hiring him would work out very well.