Parker’s last competitive game in charge of an English side was a harrowing 9-0 defeat at Anfield. A repeat never looked likely from the moment Brownhill ran clear of a young Luton defence to steer through the legs of Kaminski.
Parker has pedigree at this level - taking both Fulham and Bournemouth up - and his team were organised, confident and surprisingly direct throughout.
The first two goals came from early long passes in behind the Luton backline in a notable change from the patient possession game favoured by new Bayern Munich boss Vincent Kompany and they were a threat from set-pieces throughout.
After Chong pulled one back, O'Shea's powerful header was excellently guided in before Lyle Foster ran from his own half to tee up Vitinho for a low drive.
Twenty-eight teams have bounced back from relegation at the first attempt since the formation of the Premier League, including both Leicester and Southampton last season.
Both of these teams will aim to join that list, with Luton boss Rob Edwards saying on Friday that he “embraced” the expectation of returning the Hatters to the big time at the first time of asking.
His team has been shorn of quality in midfield and he has first-choice defenders missing but this was a worrying start.
A brief spell at the start of the second period saw Elijah Adebayo flick a header onto the bar and Chong score, but other than that Burnley were comfortable.
They won the second tier with 101 points under Kompany two seasons ago and could well be the team to beat again.