Skipper Josh Cullen and Jaidon Anthony struck the goals for the Clarets on the day.
Parker said afterwards: "It was a really pleasing performance. I thought we were really good across the 90-plus minutes.
"We started well, set about them early, limited them to maybe one big chance, but then it was all about us.
"Josh Cullen was superb - a goal and an assist, a real captain's performance. He took his goal superbly well, but his overall contribution was top quality.
"Then the pass for Jaidon's goal too was one of real quality and Jaidon was so calm, showing the nous to round the keeper and produce an excellent finish.
"That goal gave us some air really, and then we dug in, together as a collective, to keep a clean sheet which is really pleasing too."
Each player fulfilled his role
Parker added: "Each and every player, the 1-11, the subs, those who didn't get on - I can't speak any more highly of the way we all approached today's game.
"Any three points in the Premier League are massive, but this was the sort of performance and atmosphere that we will need to generate here throughout the season.
"I thought the fans were absolutely superb, they fed off the boys' performance.
"We were great value for the win and we need to continue in this vein over a consistent period now."