The Cottagers took the lead after six minutes when Ademola Lookman turned the ball into the bottom corner.
But United responded as Edinson Cavani capitalised on a mistake from Fulham goalkeeper Alphonse Areola to level before Pogba curled home from outside the area in the 65th minute to secure the three points.
“I think proud really but also disappointed, the players are also disappointed. We managed to get our noses in front and then a mistake and a screamer from a world-class player have defined the game,” Parker said.
“I think the overall factor is the big moments in the game, and I think we had the first big chance of the second half and a couple of big chances after that and they need to go in the net for us and you need to take those chances really.
“We didn’t manage to do that tonight and we probably haven’t managed to do that enough over the course of the last few games really.
“So it is certainly an area where we need to get better and we will do that.”