The parable of the wineskins is found in Matthew 9:16-17, Mark 2:21-22, and Luke 5:36-39, but only in Luke is it referred to as a parable.  A further difference is that in Luke there is an additional enigmatic sentence at the end, not found in Matthew or Mark.  For this reason we will look at the passage in Luke. 

As always, when trying to understand Scripture it is important to look at the context.  The first part of the chapter, Luke 5:1-11, describes Jesus calling Simon Peter (and by inference his brother Andrew), James, and John to be His disciples.  Later in the chapter, Luke 5:27-28, Jesus called a tax collector named Levi (Matthew) to be his disciple.  Then, in the next chapter, Luke 6:12-16, Jesus selects the twelve apostles.

So the context of the parable of the wineskins is Jesus choosing the twelve apostles.

After Jesus selected Levi, Levi held a great feast in his house, during which the scribes and Pharisees questioned both Jesus and His disciples about their behaviour.  Let’s take a closer look.

Luk 5:29  Then Levi gave Him a great feast in his own house. And there were a great number of tax collectors and others who sat down with them.
Luk 5:30  And their scribes and the Pharisees complained against His disciples, saying, “Why do You eat and drink with tax collectors and sinners?“

This is a question directed at Jesus’s disciples.

Luk 5:31  Jesus answered and said to them, “Those who are well have no need of a physician, but those who are sick.
Luk 5:32  I have not come to call the righteous, but sinners, to repentance.”

Jesus stepped in to defend His disciples.  The scribes and Pharisees thought that eating and drinking with tax collectors and sinners was not appropriate behaviour for disciples of a religious teacher.  They were in effect questioning Jesus’s choice of disciples.

Not satisfied with the answer, the scribes and Pharisees then asked Jesus a question.

Luk 5:33  Then they said to Him, “Why do the disciples of John fast often and make prayers, and likewise those of the Pharisees, but Yours eat and drink?“

The scribes and Pharisees again questioned the behaviour of Jesus’s disciples, but this time a different aspect, not who they associated with, but a seeming lack of piety.  Underlying this was really a questioning of Jesus’s choice of disciples.  The first part of Jesus’s response addresses the behaviour of His disciples and is easy to understand.

Luk 5:34  And He said to them, “Can you make the friends of the bridegroom fast while the bridegroom is with them?
Luk 5:35  But the days will come when the bridegroom will be taken away from them; then they will fast in those days.”

Jesus’s disciples were enjoying life, eating and drinking, while Jesus was with them.  Their time of fasting would come later, after Jesus’s death and resurrection.  The next part of Jesus’s response is not easy to understand, but it addresses the indirect question about Jesus’s choice of disciples.

Luk 5:36  Then He spoke a parable to them: “No one puts a piece from a new garment on an old one; otherwise the new makes a tear, and also the piece that was taken out of the new does not match the old.
Luk 5:37  And no one puts new wine into old wineskins; or else the new wine will burst the wineskins and be spilled, and the wineskins will be ruined.
Luk 5:38  But new wine must be put into new wineskins, and both are preserved.
Luk 5:39  And no one, having drunk old wine, immediately desires new; for he says, ‘The old is better.’ “

Jesus was not addressing law versus grace, nor was He addressing old teachings versus new teachings.  Firstly, this is not the context, and secondly, Jesus’s puzzling statement in verse 39, “The old is better”, makes no sense if this is the case.

Jesus was addressing the scribes and Pharisees.  They had questioned of His choice of disciples, and now He was explaining His choice by means of a short parable.

But what does this parable mean?  Everyone seemed to understand it as there is no record of further questioning.  Is there some common knowledge they had, which we do not, which made the meaning obvious?  Perhaps.  There is a well-known proverb found in the Pirkei Avot which is surprisingly similar. (The Pirkei Avot, which is literally translated as “Chapters of the Fathers”, is a compilation of teachings and proverbs from Rabbinic Jewish tradition.)  Although the Pirkei Avot is dated to after Jesus’s time, the proverbs of Pirkei Avot belong to a body of oral tradition, much of which predates Jesus.  No doubt the scribes and Pharisees of Jesus’s time were familiar with much of the content of the Pirkei Avot.  The proverb of interest is, Pirkei Avot 4:20, which states:

Rabbi Meir said: “Don’t look at the container but at that which is in it: there is a new container full of old wine, and an old container in which there is not even new wine.”

In this proverb, the vessels containing the wine are individuals, and the wine is the teaching that the individual consumes or contains.  Applying this symbolism to the parable of the wineskins we have the following.

Symbol	Meaning
Old wineskin/garment	Old students (with formal Rabbinic education)
New wineskin/garment	New students (with no formal Rabbinic education)
Old wine/patch	Old teaching
New wine/patch	New teaching

Substituting the symbols with the meaning, we can then transform the parable into plain English as follows:

No one teaches a new teaching of the Law to students educated in the traditions of the Pharisees; or else the new teaching will be rejected, and the student will be lost.  But the new teaching of the Law must be taught to new students, and both are preserved.  And no one, educated in the traditions of the Pharisees wants the new; for he says, “The old teaching is better.”

The meaning of the parable is clear: new teaching must be given to new students in order to be received.

Furthermore, the puzzling verse 39, is now clear. 

Luk 5:39  And no one, having drunk old wine, immediately desires new; for he says, ‘The old is better.’

Those educated in the traditions of the Pharisees were biased towards them.  Rejecting the new teaching they say, “The old is better”.

Summary

The scribes and Pharisees could not understand Jesus’s choice of disciples.  They lacked formal education as made clear in Acts.

Act 4:13  Now when they saw the boldness of Peter and John, and perceived that they were uneducated and untrained men, they marveled. …

At Levi’s banquet, the scribes and Pharisees criticized the behaviour of Jesus’s disciples and His choice of them.  Jesus responded with the parable, which in essence explained to them why they were not qualified for the role of disciple and why the “low-life”, with whom He chose to associate with, were.