I've never had it from sparring, but when I wake up I occasionaly have cut lips. I just put water on them and then use chapstick.
You cant accelerate healing you know. I've heard some people use vaseline, you might want to apply that to your face next time you spar. I dont think it's a bad idea.
Also it is possible your mouthpiece is leading to your lips getting cut, maybe shave the edges of your mouthpiece but it is probably just what you say in dry lips
Put vasoline on your face before sparring. Little bit on your lips, nose and cheeks will protect you from cuts. R