I have seen these things so many times. In TV serials, in my high school brief Mahabharata book and at many online resources:
Dronacharya didn't accept Karna as his disciple, as he was soot-putra. At this, Karna explains the truth, that low/high castes have nothing to do with your skills. You can learn anything and you have right to do so. Still Drona didn't agree.
Parshuram only accepted Karna as disciple when he said he is Brahmin. And at the end of his education, when the incident took place that revealed the truth, he said that a Brahmin can't have so much strength/patience, only a Kshatriya has such powers.
What is the truth? Did they have wrong beliefs? Or truth is something else?
This is so disappointing to me, as both of them were Avatars of Brahma and Vishnu. How can they believe like this?