"Does palm oil contains cholesterol?"
And, the answer is definitely "NO" .. and I'm correct.
Why does palm oil, being a vegetable oil, is cholesterol-free?
Based on what I learned in school, all the non-animal origin food, is cholesterol-free. In other words, only animal products, or any food that has animal products as at least one of its ingredients, contains cholesterol.
Next time, try to check the ingredients on packing/labeling of the cookies (or any other food product) that are labeled as "cholesterol free". You will discover that those food do not use any ingredient of animal-origin.