The first puzzle piece is on the left. To reach the top, you have to jump on the monstar falling on the left hand side, on the floor just below the cannon. This causes you to spring up to the left of the cannon, where you can easily reach the upper piece.
It's best to collect this piece after getting the key. As this puzzle piece has no green sparkle, it will return to it's original position if you move to the left.
This requires getting the key on the right-hand jump platforms. This involves properly jumping over each fireball, and reaching the top. The key has a green sparkle, and will persist once you touch it.
It's best to get the third puzzle piece to the right.
Once you have the key, fall down, and climb the left set of platforms, as if you were collecting the first puzzle piece. This lets you open the door, and collect the puzzle piece.
Opening the door from the right will not work, as it will instantly lock closed, and you can't reuse the key.
This piece is obtained in the same way you collected the key required for puzzle piece 2.