The Clarets lost 2-1 at Turf Moor in Saturday's fourth round encounter.
Parker said afterwards: “It’s bitterly disappointing. We wasted so many big chances in the first half, massive opportunities.
“And then the longer it stays at only 1-0, there’s always a chance for the opposition, and they made the most of a really below-par second half display from us.
“We were really edgy when they equalised and then went behind, but even then we had some great chances to get ourselves level.
“It just wasn’t to be and that’s hugely disappointing.”
Parker also insisted: “I think the effort was there from the boys today, but we lacked the quality when we needed it with the glorious chances we squandered.
“Just like we were praised on Wednesday for a good performance at Palace, today there’ll be criticism because of the way we played, in particular in the second half, and the result.
“We’re hugely disappointed because we saw this as a huge opportunity to build on Wednesday’s win, and get some momentum going.
“Ultimately we’ve fallen short.”