Although one must take care in drawing parallels , the appearance of parallels in Avot which contains the oldest Rabbinic traditions  would seem to be important.
In Avot 4:20:
רבי אומר, אל תסתכל בקנקן, אלא במה שיש בו: יש קנקן חדש, מלא ישן; וישן, אפילו חדש אין בו.
In Luke: 5:33-39:
33They said to him, "John's disciples often fast and pray, and so do the disciples of the Pharisees, but yours go on eating and drinking."
And no one pours new wine into old wineskins. If he does, the new wine will burst the skins, the wine will run out and the wineskins will be ruined. 38No, new wine must be poured into new wineskins. 39And no one after drinking old wine wants the new, for he says, 'The old is better.'
It would seem that the use of this particular parable, in the context of a polemic against the Pharisees, using a phrase that we know was adopted by the Pharisees, is particularly significant. But I am not certain how to interpret this? Ideas, anyone.
[I thought of one possibility. Luke was the companion to Paul who is most famous for propagating the the abrogation of Jewish law (satirized in BT Shabbos - Ch. 16) [Since as Kevin notes, it also appears in Mathhew - this is nonesense].
This interpretation depends on the (admittedly questionable) assumption that the Wine parable was current in the circles of the Pharisees before Rebbi. An alternative explanation would have Rebbi differentiating between himself and the Christians by stressing that he is merely changing the form of the law, not the law itself. But the expression isn't formulated correctly for such a polemic.
[Note that the parable of wine and its vessel is also used in the argument between R' Yehoshua ben Chananya and the princess - Taanis 7a]
[1a] The credit for the idea behind that post goes to my friend, SL Rubinstein.
 For instance one cannot deduce anything from the "eye of a needle" parallel as this is likely a popular expression that was in wide use.
 See R' Dovid Tzvi Hoffmann "HaMishna Rishonah". I will discuss this sefer when I continue my series on him at some future time.
According to some commentaries, the connection between this phrase and Rebbi is that Rebbi as redactor of the Mishna was pointing out that despite the "new" form of his creation it is still full of "old"er traditions. This perhaps has important implications regarding the question of Rebbis actions as redactor of the Mishna. Perhaps this is proof that the original form of the traditions were as a Midrash and Mishna is a new form originated by Rebbi. [I erred in ignoring the context in which Rebbi is clearly responding to R' Yosi B'R' Yehuda who says there is no point in learning from a young person (as it is used by Baal HaMeor in his intro). (I am reminded of the Gemara in Bava Metzia (Ch. 4) where Rebbi says - "ילדות היתה בי והעזתי את פני ר' נתן הבבלי" - i.e. he is admitting the deficiencies of youth.]
It is clear that Rebbi didn't originate this phrase since it appears aleady in Luke, before Rebbi was born.
[Update: Potentially interesting book (Google books will only let me see so much), some of his claims in the page linked are invalid. Jesus's anti-divorce sentiment may reflect the opinion of Beis Shammai (see Rosen-Tzvi's article in JSIJ), the fasting is the subject of my discussion here but is not simple (if he believed himself Messiah then, like the later Shabsai Tzvi, he would naturally believe fasts need be abolished - "Can you make the wedding guests fast while the bridegroom is with them " - note the parallel to Mishna Sukkah where guests at a wedding are also exempt from Sukkah).]
 But if the thesis in this book is correct my whole argument has no basis. I don't know waht to do with the preceding parable about clothes and with the concluding statement that "old wine is better".
Perhaps: "The old is good: this saying is meant to be ironic and offers an explanation for the rejection by some of the new wine that Jesus offers: satisfaction with old forms will prevent one from sampling the new." (here) but this is difficult since it is well-know that old is better (and this can be seen in both the Bible (Esther), and Rabbinic documents). This is clearly a Paulian apologetic