When I create a new object (for instance: cube or cylinder), I can change it's vertex via the parametric menu, but when I rotate or move the object, I lose the parametric menu! My question is: can I keep the parametric menu active even after I move or rotate the object?
The 'parametric' menu is actually the new objects menu. It is only available when you create a new object. Therefore there is no way to keep it after you have moved, rotated, scaled, or edit your model in any way. The reason for this is that when you do any of those things, your model is considered edited and therefore unable to change its own vertices. (You have to change them from edit mode)
I hope this helps.