I’ve heard the explanations that say that Jesus was talking about a weed that looks exactly like wheat when it is young, but becomes more obvious later in the season. That’s an interesting explanation, but it’s not explicit in the parable itself (although it might be an assumption Jesus is making, based on his audience).
However, parables aren’t meant to be full and complete descriptions – rather, they illustrate one facet of the lesson that’s being taught. So, I think it may be overly ambitious to ask the question “can weeds turn into wheat?”. It seems that this parable addresses other questions: how does evil come into the world? why does it spring up all around us? why doesn’t God root it out as soon as it’s present?