The 1-0 reverse confirmed Burnley's relegation.
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“I’m immensely proud of the team tonight," Parker said afterwards. “It was a formidable challenge and going a goal down so early on could’ve easily derailed us, but we stuck to the task.
“We created chances, which we weren’t deadly enough to take, and at the other end we defended well.
“We’ve left everything on the pitch tonight.”
On relegation, Parker added: “The curtain has come down on us now. We’ve failed and we’ve fallen short.
“The team has given me everything this season - but at times we’ve lacked the quality required in this league.
“We’ve not managed to survive. We’ve face a lot of adversity. But we’ve come up short.
“We need to learn from our mistakes as a football club.”