The Clarets were comfortably beaten 2-0 by Everton through goals from James Tarkowski and Kiernan Dewsbury-Hall either side of halftime.
Parker conceded afterwards: “We fell short in many aspects of the game tonight. Partly down to their quality - but to be honest, we were never in the game from start to finish. We were second best and never really laid a glove on them.
“It’s a disappointing night for us.”
He also said: “We’ll debrief tonight and in the coming days. That’ll be important for us.
“We’ll need to look back in more detail as to why we produced that performance tonight.
“There’ll be a lot of searching for answers over the next few hours and days.”
On the traveling support, Parker added: “I understand the fans’ frustrations. We gave them nothing to shout about tonight.
“It was a poor performance - maybe the weekend took its toll on us.
“We’re lacking in consistency at the moment - that’s disappointing and has to change if we’re to get ourselves out of this position we find ourselves in.”