Does a parent class always need a default or no-argument constructor even if the child class already has a defined constructor?

Am I right to conclude that a parent class always needs a default or no-arg constructor?

No. The first thing a subclass constructor must do is to invoke one of the superclass constructors. If you don't, then the compiler calls the no-arg constructor of the superclass for you. But of course that fails if the superclass doesn't have one.

Your code should be:

class Pony extends Horse {
    Pony(int age) {
        super(age);
    }
}

See, the superclass already has an age field, and probably methods that use that field. So redeclaring one in the subclass is wrong and counter-productive.

To make it compile to you need to call super(int) instead of calling super() implicitly.

Pony(int age) {
    super(age);
    this.age = age;
}

Then you may notice that both Horse and Pony have exactly the same field which holds exactly the same data. Why would Pony need to have the field age if the class Pony is based on (Horse) already defined this field?

Pony(int age) {
    super(age);
}

It's the beauty of inheritance: you may make use of anything that was already written in the base class.

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However, the compiler complains that the parent class has no default constructor.

It complains because, as I mentioned above, it failed to invoke super().

If a constructor body does not begin with an explicit constructor invocation and the constructor being declared is not part of the primordial class Object, then the constructor body implicitly begins with a superclass constructor invocation "super();", an invocation of the constructor of its direct superclass that takes no arguments.

^{https://docs.oracle.com/javase/specs/jls/se12/html/jls-8.html#jls-8.8.7}

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What if I want the parent and child class to only have constructors with parameters?

Then the child class should always explicitly state what parent's constructor it wants to call.

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Am I right to conclude that a parent class always needs a default or no-arg constructor?

No, it's fine to have a parent class without the default constructor as long as its children don't invoke super().