I just bought Parappa and played through the campaign in one sitting, on the default difficulty, and I didn't have any real problems except for being able to confirm what everyone says about the timing being wrong in stage 4. It took me about half a dozen tries to beat stage 4, but I finally finished it with a Good! after realizing that, only in that stage, you have to hit the buttons so Parappa says the words just a fraction of a second too early. Weird. I don't remember if it was that way in the PS1 or PSP versions, but that stage does feel a bit broken. The timing felt pretty good on the rest of the stages, though, or at least as good as it was on the ps1/psp.
Yep, the original was like that, too. I got the original on the first release and loved it up until stage 4. You can ask my college roommates just how much I flipped out during that stage every time I lost. The PSP version was also like that I think. I would beat the game many times overall, but it's disappointing to read that this remaster didn't at least fix this stage.
Still, I may buy it during the flash sale despite how the tap timing has always been off. It's still so hard not to love the utterly lovable characters, songs, and presentation after all these years.