The parable is a series of contrasts between new and old. It contains new and old clothing, new and old wineskins, and new and old wine. Christ's being taken away makes the “newness” possible, and once that “newness” is available, it is wholly incompatible with the old.
Jesus begins with an example of old and new garments: “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.” In Scripture, going all the way back to the Garden of Eden, garments or clothing are common symbols of righteousness. After Adam and Eve sinned, they tried to cover themselves with something they made with their own hands (Genesis 3:7). Instead, God gave them tunics made of skin (verse 21), requiring the life of an animal, representing the Lamb of God giving His life to cover sin.
Matthew 22:1-13 contains the Parable of the Wedding Garment, whose lesson is that inappropriate clothing will keep a person out of a wedding feast. Isaiah 64:6 says that “all our righteousnesses are like filthy rags.” The Pharisees had a righteousness, but Jesus asserts that our righteousness must exceed theirs (Matthew 5:20), meaning that we need to have His righteousness imputed to us, which becomes our new covering, our new garment. As we become one with Him and submit to taking on His image, we have a righteousness that does not come from our works but from God's work in us.
Thus, we have a contrast between man's righteousness and the righteousness of Christ. But, just as it makes no sense to tear off a piece from a new garment to patch an old one, so is it also a futile exercise to try to keep our own righteousness intact and use a little bit of Christ's righteousness to cover a flaw here and there. The two coverings are incompatible—we have to choose one or the other.
The conclusion is that, if a new garment is available, we would be foolish to use it to mend an old, defective one. Because Jesus was taken away, His righteousness is available to us, so we need to discard any thought that our own is suitable. Instead, we must put on His righteousness and be conformed to it so that it fits and covers us appropriately. Clearly, works are involved and required on our part, but without the covering and involvement of Christ, those works would continue to be as filthy rags.
To understand the new and the old, it is important to realize that the “old” could have many applications. It is not just the Old Covenant. In fact, the Pharisees in Jesus' audience did not actually represent the Old Covenant. The system of beliefs and practices that developed into Judaism is not the same thing as the Old Covenant. Certainly, Judaism makes use of the writings of Moses and the prophets, but it also leans heavily on the traditions of Jewish scholars and is infused with Greek philosophy.
The Pharisees, then, were not actually living by the Old Covenant! God intended that covenant to prepare His people for the coming of the Messiah. Everything in the holiness code, the sacrifices, and so forth was intended to point to Christ. Since the Pharisees could not recognize the Object of the Covenant, what they were practicing was not what the pre-incarnate Christ delivered to Moses. They had gotten far off course.
Therefore, the “old” elements in this parable could be any system of belief aside from what became available through Christ.