Does an airfoil drag coefficient takes parasite drag into account?
Airfoil drag is &quot;parasitic&quot; (or better: everything but induced) drag.
It consists of shear drag and pressure drag, the latter mostly from local flow separation. Both are only present when viscous flow is assumed. Airfoil drag is for the wing section without taking tip effects into account, presuming an infinitely wide wing. This kind of theoretical wing has no induced drag (d'Alembert's paradox).
L/D max is the point on the polar curve where the angle to the origin of the coordinate system is steepest. This is not the point of the lowest drag coefficient! Maybe you mean the point of lowest drag for a full airplane – but that is another story, with more than just the airfoil polar. Now induced drag is part of overall drag and due to its inverse relationship with speed, a distinct minimum can be found when lift is held constant.

Drag polar of the NACA 23012 airfoil (picture source, colored comments own work)