How can I make a dynamic array of pointers that each point to an abstract base class?

Your thinking is correct, but your implementation of add() is not optimal. If nothing else, it is leaking the old pets array. But it is also just coded more complicated than it needs to be.

It should look more like the following instead (assuming pets and length were initialized properly before add() is called, and that PetArray is managing them correctly per the Rule of 3/5/0, and that your PetArray object is not being corrupted from another bug elsewhere in your code):

void PetArray::add(Pet *p)
{
    Pet** temp = new Pet*[length+1];
    for(int i = 0; i < length; ++i)
    {
        temp[i] = pets[i];
    }
    temp[length] = p;
    
    delete[] pets;
    pets = temp;
    ++length;
}

UPDATE: you are not add()'ing objects to your array correctly. You are creating your derived objects in automatic memory, not in dynamic memory. You are taking pointers to the objects right before they go out of scope and get destroyed, and then you add the now-invalid pointers to your array.

Pet* p;
...
if(species == "Cat"){
    ...
    Cat c(name, species, age, weight, length);
    p = &c;
} // <-- c is destroyed here!
// same goes for the other animal types, too...
...
add(p); // <-- p is invalid here!

This explains both problems you described. You are adding pointers to local memory that gets reused in each loop iteration, and then you crash when trying to access invalid objects.

You need to new the objects instead, eg:

Pet* p;
...
if(species == "Cat"){
    ...
    p = new Cat(name, species, age, weight, length);
}
// same for the other animal types, too...
...
add(p);