Calling the base class constructor from the derived class constructor

First off, a PetStore is not a farm.

Let's get past this though. You actually don't need access to the private members, you have everything you need in the public interface:

Animal_* getAnimal_(int i);
void addAnimal_(Animal_* newAnimal);

These are the methods you're given access to and these are the ones you should use.

I mean I did this Inheritance so I can add animals to my PetStore but now since sizeF is private how can I do that ??

Simple, you call addAnimal. It's public and it also increments sizeF.

Also, note that

PetStore()
{
 idF=0;
};

is equivalent to

PetStore() : Farm()
{
 idF=0;
};

i.e. the base constructor is called, base members are initialized.

The constructor of PetStore will call a constructor of Farm; there's no way you can prevent it. If you do nothing (as you've done), it will call the default constructor (Farm()); if you need to pass arguments, you'll have to specify the base class in the initializer list:

PetStore::PetStore()
    : Farm( neededArgument )
    , idF( 0 )
{
}

(Similarly, the constructor of PetStore will initialize sizeF, by calling the constructor of Farm. The constructor of a class always calls the constructors of all of its base classes and all of its members.)

The base-class constructor is already automatically called by your derived-class constructor. In C++, if the base class has a default constructor (takes no arguments, can be auto-generated by the compiler!), and the derived-class constructor does not invoke another base-class constructor in its initialisation list, the default constructor will be called. I.e. your code is equivalent to:

class PetStore: public Farm
{
public :
    PetStore()
    : Farm()     // <---- Call base-class constructor in initialision list
    {
     idF=0;
    };
private:
    int idF;
    string nameF;
}