There is no doubt that learning from mistakes is an important experience during one's development of personality. I can't imagine what my life will be like if I didn't make any mistakes in my childhood. For parents, they should allow their children to make mistakes and let them learn from their own mistakes.
Many parents hold the view that, especially in Chinese conventional concepts, the experiences that the elders gained from mistakes is the most precious gift can be given to the younger generation, and children's life may be easier with less mistakes if these experiences can be used for reference. Though this kind of notions make sense in a way, actually, it's better to learn from one's own experience, which can't be replaced by others'. It is generally believed that people tend to be more impressed by what they have done themselves, for the reason that others's experience is hard to be 100% accepted and learned from.
In addition, children might learn more from their own mistakes, at least more than solely listening to parents, with their open-minded ways of thinking. We can observe that those who always got in trouble during childhood tend to have better ability of dealing with problems when grow up.
To be further, it's not wise for parents to impose ideas to children. In other words, children have the rights to live in their own ways, I mean, their own way to feel and think, including make mistakes. Some might argue that kids don't have whole personalities and so need parents to lead them to a right way, but it doesn't mean their own mistakes is unnecessary.
In a word, I believe that an excellent leading role of parents is to show the children how to think and conclude something from their mistakes, but not preventing them from making mistakes.