Is there any textual evidence to suggest that the explanations to Jesus' parables such as the explanation given to the disciples regarding the Parable of the Sower in Matthew 13:18-23 were not Jesus' own words?
I'm going to focus on just the explanation of the parable of the sower and not (for the moment) consider other explanations. Conveniently, that text is found in all three Synoptic gospels. Assuming Mark was the first written, we can get a pretty good idea of how Matthew handled the material:
When you look at the diffs, the changes are:
There are a few word order, punctuation, and style changes, but they don't really have a huge impact on the meaning. If you do the same exercise with Mark and Luke 8:11-15, you've find similar results. Neither accepts Mark's wording as "gospel truth", but neither starts over from scratch either. It comes as no surprise to careful readers that Matthew inserts the idea of the kingdom into the text. He has a slight aversion to the word Satan it seems. These changes are of focus, not of fact.
Therefore, we can be certain that neither Matthew nor Luke are the source of this particular teaching. But that merely pushes back the question: did Jesus or Mark orginate this particular explanation of the parable?
Unfortunately, the text itself doesn't help us here. Or at least, the text clearly puts these words in Jesus' mouth, but it can't prove to a critic of the text that it wasn't added to the tradition sometime after his crucifixion. It's fair to say that if Matthew is really Matthew/Levi, one of Jesus' chosen twelve, and if Luke talked to some eyewitnesses about this particular section of Mark, then the words are from Jesus himself. Why else would these witnesses accept Mark's text more or less intact unless it was more or less how they remembered events occurring?
However, to a scholar who is skeptical that Jesus' original followers had a hand in any of the gospels, the above analysis is pretty good evidence that Matthew and Luke simply accepted and tailored Mark's account without verifying it. And if Mark were separated from Jesus by several steps, he might have confused the commentary on Jesus' teaching with the teaching itself. Matthew 13:10-17, which explains why Jesus taught in parables and why most people didn't understand him, sounds like a plausible post hoc justification of the unpopularity of Christianity among the Jews of the 1st century.
Personally, texts like this one confirm to me that the gospels accurately recount the teaching and life of Jesus. This passage follows a section particularly critical of the folks who listened to Jesus teach. All four canonical accounts (Mark especially) relentlessly portray the twelve disciples as clueless. Matthew and Mark recount how immediately after Peter confessed Jesus as the Messiah, Jesus calls him Satan for failing to understand that the Son of Man must suffer.
Later accounts, the non-canonical Gnostic works in particular, tend to elevate one or more of the disciples as particularly aware of what Jesus was trying to say. The idea that the twelve just didn't get it must have been strange to early observers of Christianity and one solution might have been to retcon the narrative. That the canonical gospels refrained indicates that they are historically reliable.
My analysis suggests that Jesus sometimes explained his teaching to his followers privately and some of those explanations made it into the text of the Synoptic gospels.