Despite snowy conditions in north London the game went ahead, and Dempsey netted in the 93rd minute to ruin United's hopes of going seven points clear of Manchester City after Robin van Persie's first-half header.
Both Parker and the goal-scoring hero feel Spurs can now build on this performance and work towards qualifying for the Champions League after they strengthened their grip on fourth place.
"We came here and really if there was ever a time we were planning on getting a result it was today," he told Sky Sports. "I don't think Man United were probably at their best and we put a lot of pressure on them in the second half and we got the deserved goal. We deserved to get a point out of that game, definitely.
"I think we've shown a real resilience, we've had a good first half to the season. We need to now go and push on, keep getting points, keep as tight as we can. Our objective is fourth but if we can improve on that and get better than that then so be it."
Dempsey has now scored twice against United this season having previously failed to find the target against Sir Alex Ferguson's team and was hopeful of Spurs' chances of remaining in the top four.
"We believe in ourselves and we've been getting good results and been on a good run of form," he said in the same interview with Parker. "I think we have the quality in ourselves that we can get results that will keep us in the top four."
The American forward also expressed astonishment at David de Gea's superb reaction stop to deny him in the second half.
"Oh man, he's a top goalkeeper and he made good saves but I couldn't believe he saved that. I though it was going in and I was going to celebrate, what a great save," Dempsey remarked.