Both are correct. In my experience, there is no significant difference in meaning.
I will say that I personally feel like "He didn't dare contradict his parents" sounds more educated, but that's entirely a subjective assessment, and others may feel differently. But neither sounds wrong.
Note: When dare is used in the imperative (command) form, it is almost always used with to. So "Dare to try!" will be much more readily understood than "Dare try!" (and some may argue the latter is even incorrect).