Can a 13 year old defend themselves in court without a lawyer?

Not that I'm aware of. And honestly not a good idea for anyone to represent themselves if they are not familiar with the rules of the court and procedures. They could end up paying fines, the other side's court fees or held in contempt.

*Parents cannot either because they do not have license to practice law. If you need an attorney but can't afford one, you'll be appointed a public defender.

No, a 13 year old can't defend themselves. In the U.S. and many other western countries, you need to be at least 18.

While an adult has the right to do so no judge would allow a minor much less a 13 year old to do so.

A person who represents themself in court has a fool for a client. NEVER NEVER go that route unless it's a traffic ticket. Certainly not a minor, no.

Some people do it successfully. Others do a miserable job of defending themselves or even proving a case as a plaintiff and they lose. The answer to your question is, it depends on the competency of the person and how well they understand the law.

Their parents would pay for a lawyer for a minor.

The constitution says they can. Repping yourself never works out.