Jaidon Anthony hit Burnley's equaliser on 66 minutes after United had jumped 2-1 ahead through a Benjamin Sesko double.
Parker said afterwards: "I was really pleased with the performance and the reaction of the team. We demanded a reaction, all of us, and we got that - in two ways. With our performance in general, but also recovering from going 2-1 down. That was great to see. It was a good reaction and a good point for us.
"We defended for our lives at times. None of my players left anything out on that pitch. That fighting spirit was key tonight. We went to places that are very extreme. We brought energy and fight - and deserved what we got from the performance."
On Anthony's goal, he continued: "It was a brilliant equaliser. A brilliant finish.
"It was an outstanding goal and Jaidon’s overall display from the bench was very positive. He brought an energy and a calmness to proceedings. I'm really pleased for him. That’s exactly the impact you want from players coming on."
The result leaves Burnley eight points from safety.
But Parker insists: "There’s belief here. We’ve had tough times of late but we had plenty about us tonight.
"That is what we’re going to need for the remainder of the season. This is the toughest league in the world. We understand that. We’ll keep learning and keep progressing.
"We need to keep going and keep moving forward, together."