Bonus rounds are hosted by Master Onion and happen after a level where you did fairly well, I believe. Either that, or it happens every 2 levels like it did for me.
Your objective is to hit tiles held up by Onion's disciples and the points are accrued by how many tiles you hit.
The trick to get a series of 40 is to take your time and let the attendants setup the pieces for you Parappa to break.
The tiles' positions would match the position and color of your Playstation controller buttons.
green tile (up) - triangle
red tile (right) - circle
pink tile (left) - square
blue tile (down) - x
When you see a tile ready, push the corresponding button to break it.
I usually wait for at least 2 tiles to be ready and then press their buttons quickly in succession. (Parappa's animation as he hits would make spotting on the fly much harder, so it's best to memorize what you want to hit first.)
The hardest one to spot for me was the blue tile on the bottom, since it was laid down on a stone rack instead of being held up.
As long as you don't get ahead of yourself and try to hit an empty spot, you'll be okay.
I was able to get this about halfway through my session.