Rocket Knight

As stated, you need to finish every level of the game under the par time. The time limits for each level are listed below:

Zephyrus
1.1 - Countryside - 6:00
1.2 - Tree Tops - 5:30
1.3 - Zephyrus City - 7:00
1.4 - Forest Shredder - 3:00

Wolf Kingdom
2.1 - The Counter Attack - 7:30
2.2 - Airship Armada - 4:30
2.3 - Wolf Fortress - 7:30
2.4 - The Wolf King - 3:30

Deep Underground
3.1 - Pig Bunker - 8:30
3.2 - Cyberswine Labs - 9:00
3.3 Axel Gear - 4:30

Return to Zephyrus
4.1 - The Pig Fleet - 4:30
4.2 - Enslaved City - 8:00
4.3 - Final Boss! - 5:00

The time is displayed at the top of the screen at all times, so if you're taking too long, just go to the pause menu and hit restart to try again, this won't cost you any lives. I actually didn't have too many problems with this once I was a little more familiar with the game and it's mechanics. My advice would be to utilise the rocket pack boosts to clear the areas quicker and also dodge/blast through enemies. Don't bother trying to kill them if you can just bypass them, and also don't bother about power-ups and gems so much. The only ones I picked up were a few health boosts when I needed them, otherwise I just concentrated on speed-running the level.

The levels are fairly linear, so it's not really a case of finding a secret shortcut or anything, just merely clearing it as quickly as possible. Also, it would appear this doesn't need to be done in one sitting, when I attempted this the achievement unlocked for me on the second to last level, so I can only assume that I had already beaten the par time on the last level.

Good luck!