Should I use parchment paper on a non stick cookie sheet?

It's not necessary but it makes cleanup even easier and extends the life of the non stick coating. If your non stick sheet is new and in good condition, you can bake directly on it. As the coating ages, parchment provides backup insurance. For high sugar cookies that tend to caramelize and stick, parchment is always recommended regardless of the pan's coating. Silicone baking mats are another excellent alternative.