Years ago, I tried a beginner's boxing class in college. One of the moves taught was parry an incoming jab with your back hand. This is for when both partners are in orthodox position. I think there may have been a counterjab at the same time. Since it was a beginner class, we didn't learn slipping. I picked that up later in kickboxing.
I thought, why not synchronize all three? The parry assists the slip (a bit) -- as you move your right hand leftward toward the centre line, it can help you move your upper torso rightward away from the centre line. This moves your left shoulder toward the centre line, a bit to the left of the incoming jab, and well positioned to unleash a jab of its own.
I haven't trained in years, but I try to keep up the drills in isolation. It's hard to know how practical the above drill is in a live interactive situation, even in a friendly drill, much less an unchoreagraphed situation. Can anyone comment on this? I haven't found examples online, and I'm afraid that this means it's not all that practical.
I also have a similar drill for an incoming cross, but with everything in mirror image, i.e., leading left hand parries, upper torso rotates a bit to the left, and the simultaneous counter is my own right cross.
For both of these, I sometimes lean or shift forward a bit, and sometimes backward a bit. There are, of course, follow-up strikes to the simultaneous counter, which can vary depending on one's creativity.
What I found about parrying on YouTube
Only the final video shows a slip-ish motion, specifically accompanying a side parry. To me, this makes sense because slipping wouldn't really complement the other parries that are demonstrated. The demonstrator didn't actually say to slip at the same time, but you can see him doing it.
The demonstrator only talks about a side parry against a cross, so there is no footage of him parrying a jab. I'm wondering whether slipping also makes sense for the latter.
I have yet to encounter a video on simultaneous parrying, slipping, and counter-striking.