I would like to know if I can use parchment paper for any kind of cookie?
The short answer is yes. Parchment cannot be used as muffin/cupcake paper, (this actually goes without saying).
Parchment paper can be used for virtually anything, unless it is extremely oily and/or in a near liquid state.
Parchment paper typically lists a maximum temperature of 400°F. If you start getting above that temperature (450-475°F), you'll start to see browning around the edges and it's crumbly after coming out of the oven.
Parchment that's close to the food is generally safe, because the moist dough itself can't get above boiling. In the cases where you have something that isn't moist (eg, drying out biscotti), you typically do using low heat.
I'm not aware of any cookie recipies that would be a problem, but if you had one that called for high-heat cooking, especially if the batter is low-moisture, I'd avoid using parchment, particularly for that last tray where you don't have the tray well-covered with cookies.