Axel Tuanzebe and Lyle Foster struck for Burnley, with Micky van de Ven and Cristian Romero - on 90 minutes - scoring for the visitors.
Parker said afterwards: "I thought we did enough to win the game, so we’re all hugely disappointed to concede an equaliser so late in.
"You always understand that the dynamics of football mean they’ll commit extra players forward at that stage of the game - and it was one of those that came up with their equaliser in the dying stages.
"We’re all really disappointed we couldn’t hold on after another really good performance from the team.
"When I put my head on the pillow tonight, I’ll do so reflecting on more positives and a performance we can be proud of."
Parker also hailed the Turf Moor support on Saturday.
He added, "It was a brilliant atmosphere today. The best atmosphere of the season so far.
"I thought the fans were superb and really spurred us on. We were all in it together and that makes us stronger."