AlienCruise

This achievement is actually much simpler than it sounds, because the requirements are not as strict as the description makes it seem. The stage referenced is the third stage of the Star of Parcaneas, which is the big pink planet in the corner. However, you do not need to complete the entire third stage without taking damage, despite the description implying that.

During the third stage (which begins after destroying the midboss in the second stage) you'll be introduced to the volcano-like parts that shoot flames out. After some scrolling to the right, you'll head diagonally downward to a lower level. As you head right, you'll come to volcanoes on the top and bottom of the screen that alternate erupting. It is this specific section of the third stage that you need to complete without taking damage. The volcanoes spew a bit of flames to indicate they're about to erupt, so they are very easy to avoid. Unfortunately you can't shoot through the flames to hit the enemies, but there aren't many enemies to shoot. Just stay alert, and this section only lasts for about 30 seconds, making it very simple to get through without taking damage.

Unfortunately, if you do take damage, you'll need to start the level over from the beginning and get back to the third stage to try again, but this only takes around five minutes or so to do.