Daniel Munoz struck in the first-half for Palace's 1-0 win, with Jacob Bruun Larsen hitting the crossbar for Burnley late on.
"We should have definitely got something out of the game," said Parker afterwards. "I’m weak in my position when I say that, because we’ve not got anything to show for our efforts again.
"But I asked the team to be brave and fearless and show our quality, and they did that. I asked the lads to take away the noise - because the world we live in, there’s lots of noise these days.
"It didn’t fall for us tonight. I keep talking about fine margins, that’s the Premier League for you. It’s frustrating and disappointing."
Parker added: "This game kind of sums up lots of matches we’ve been involved in during the last few weeks. The goal came at a horrible time for us. It was a big blow just before half time.
"We’re not getting the breaks at the moment but this group will not wilt. What’s key now is keeping stable. Of course we need to turn the tide, but I’m seeing a lot of things that encourage me. I’m proud of the group tonight."
On his substitutes, Parker added: "I thought they had a massive impact. That really pleased me and represents what this team stands for in terms of the togetherness of the group.
"We're fighters, who fight for each other and the badge - we'll keep going."