Can we call a parent function like this? this doesn't work it said "function cool no exist"...somebody else?

/////////////////////////////////////////////////////MAIN
myClass ismyClass;

void setup(){
 size(300,300);
 ismyClass = new myClass(this);
}

void draw(){
 this.rect(20,20,20,20);
 ismyClass.display();
}

public void cool(){
 rect(150,40,34,344);
}


/////////////////////////////////////////////////////CLASS MYCLASS

public class myClass {
 PApplet parent; // The parent PApplet that we will render ourselves onto

 myClass(PApplet p) {
   this.parent = p;
 }

 // Draw stripe
 void display() {
   parent.background(10);
   parent.cool();
 }
}

///////////////////////////THANK SO MUNCH
and happy new year and best wishes for all!

parent is defined as a PApplet, so the compiler knows only the methods in PApplet, coming with Processing.
cool() is part of your program, which is actually inside a class created by Processing, of the name of your sketch, extending PApplet.
Something like:
Code:

class MyFabulousSketch extends PApplet {
  void setup() {} // Override PApplet's setup
  void cool() {} // Add a new method not in PApplet
} 

Since myClass is actually a nested class inside MyFabulousSketch, as blindfish wrote, it can access all the methods of the parent without problem.
If you really want to use parent, you can do it in a convoluted way (untested):
((MyFabulousSketch) parent).cool()
ie. you tell the compiler that parent is more than a PApplet.

Libraries are another story: their classes are not nested in a PApplet, so they don't have a direct access to its methods, hence the need to pass a PApplet instance to them, eg. to draw on the surface of the sketch.
That's what you would get by putting myClass in a .java tab.

In short, for classes in .pde files (main or secondary), you don't have to pass around the 'this', it is always available.