Does Blender allow to perform operations on multiple objects at once? Let's say five objects are parented to the same 6th object. Is it possible to remove the parenting links at once. Selecting all children and removing the parent doesn't work:

  

Only the "active" object is un-parented.

That's curious since the inverse operation, parenting multiple objects to the same parent, is possible at once.
You are not doing inverse operation to parenting. Modifying a parameter of active object (properties display active object) only influences that object.

Operators can operate on selections, like Ctrl+P and Alt+P do.

If there is no operator to do it on multiple objects then RMB on the field > Copy to Selected. Works on most parameters in general.