Cuphead

Well here's me doing the pretty much the same thing as New Paralyzer, I thought that I was first

In Inkwell Isle II (World 2), you'll want to go to the Run 'N Gun level Funfair Fever, located because Baroness Von Bon Bon. Near the beginning of the level, you'll notice a trampoline with some high walls and balloons. Some of the balloons are pink and can be parried! As a pink balloon comes across, begin to parry and follow the balloon to get 6, 7, or even more parries in a row. Also, after this, go see the juggler in World 2 to be given a extra coin!

There may be earlier opportunities to get this, but Mausoleum III (the third ghost defense mini-game) available in the center of Inkwell Island Three, features multiple chances to get 5 parries pretty naturally. (It may be difficult to complete this mausoleum without getting this.)

A new type of ghost is introduced this time that splits into two when hit, effectively giving you three parries right off the bat. Couple this with another two nearby ghosts and you're good to go.

The Mausoleum is accessible from the top of the map (after defeating the Theater boss) or the right (after beating Perilous Piers run & gun).

The earliest place I've found to do this is on the second island. You'll have to beat the "Sugarland Shimmy" boss, Baroness von Bon Bon, in order to open up the path above it. This can be done on "Simple" difficulty.

Just beyond Baroness von Bon Bon is a run-and-gun level called "Funfair Fever." Start it (on "Simple" difficulty if you wish), and run past the first enemy to the trampoline. Use the trampoline to jump up on the dividing wall between here and the next spot, and wait for a pink balloon to float by. (You'll need to avoid the non-pink balloons while you wait.)

Jump onto the pink balloon, and time your next jump button press for a parry as you hit it (or just before, that seems to work best for me). You'll need to time five in a row; the achievement will pop right after you do it. It's not the easiest thing in the world to time, but it's easier than most of the other stuff in this game. Good luck!

EDIT: Chui points out that you can equip the P. Sugar charm to auto-parry the first of your five.

Here is the way that I completed this achievement and it was super easy!

Hands down easiest way to do this is on World 3 during the Honeycomb boss. Play it until you get to Stage 2 and eventually the queen will shoot out giant purple circles. You may have to reset a couple of times to see this. Anyway, once you see the purple circle, jump in it and simply mash the A button. (I managed to get like 20 parries in a row, lol)