The main teaching methodology used by Jesus to the masses was parables. Symbolic and analogous stories with deeper spiritual meaning. It is interesting when we get to the time of the Apostles that we do not find them using the same method. Instead we find them in more direct proclamation and apologetics. Why was this?
I think the first thing to say is we actually don’t have much evidence of actually how the apostles taught. After the Acts of the Apostles we enter into different literary genres (letters and apocalyptic prophecy). However we do have the Acts of the Apostles. It is helpful that both Luke and Acts are written by the same author as we have a direct comparison. The reason that we have no parables recounted in Acts is not simply because of a different author.
Before I get to my hypothesis as to why we find no parables, I just want to make a point that I think is important in New Testament exegesis. There has been an approach to this subject that I believe drinks too heavily from 19th century higher biblical criticism, and what I mean is this, we examine each piece of literature in the New Testament independent of the other parts and as such we seek (for some consciously, for others unconsciously) to drive a wedge between Jesus and the Apostles (in particular Paul), both in terms of message and methodology. I believe the greater joy and profitability in New Testament exegesis is to mine areas of synergy. We all know the individuals were different, with varied educations and experience, but as believers we should believe in the divine inspiration of scripture. To seek out the divine is more profitable than not, after all isn’t that why we follow Jesus?
With this in mind I think it is important to put ourselves in the mindset of the early apostles. They had been given the Great Commission from their master. A command to make disciples, teaching them everything he taught. My hypothesis (and I know it is largely from silence) is that the disciples did this. They taught what Jesus had taught them; meaning that they told parables…. Jesus parables. The other parts of Acts and indeed the letters are simply commentary on the teaching of Jesus. If this was the case then the New Testament writers (Luke in particular) would find no need to repeat what had already been written. However it was important to gather some of the Apostolic commentary on Jesus teaching which would add to our understanding of the teaching of the master.
It is also very probable in my opinion that the Apostles told their own parables, which were perhaps not directly building on Jesus points and therefore were not included. So there you have it… I believe the apostles did tell parables.