Is it permissible to sell one's personal gold and donate its value to the mujahidin or poor Muslims without the parents' knowledge, and is this gold considered private property that one may dispose of?

A gift is established by explicit wording or by indicative circumstantial evidence. A gift is also valid through mutual consent if it indicates the gift. If the parents explicitly gift gold to their daughter, or if there is circumstantial evidence of such a gift, she has the right to dispose of it if she is an adult of sound judgment, without requiring her parents' knowledge or permission. However, if you doubt their intention, it is not permissible to dispose of it without their permission. And Allah rewards the servant for the sincerity of his intention if an impediment prevents him from acting.