What these parallels mean exactly is the real story in my opinion. If you want these parallels to support a case for an either/or scenario - either Thomas is dependent or independent - you probably won't like what I have to say. I have a full discussion of this on pages 15-24 of The Original Gospel of Thomas In Translation, 2006. The issue is very complex, and suggests a little of both. I think the most likely scenario (that makes sense of all the evidence) is that as time went on, versions of the older, independent sayings came to be influenced by memories of the sayings in other gospels, particularly Luke. Also it is quite likely that there were later sayings added to the Gospel of Thomas as it developed as a text, accretions based on the gospels as I think may be the case with sayings 3 and 113 (//Luke 17:20-21). Was Luke a favorite gospel in eastern Syria? I had a private conversation with Professor Vernon Robbins at the SBL meeting in Washington, D.C., and he is working on this question right now. I am really looking forward to reading about his findings because, from what he told me, I think they will go a long way to help us sort this question out.
Another factor we have to take into consideration is that the ancient world functioned within the parameters of an oral consciousness. This means that written texts weren't actually read by all that many people. People would hear things and then rely on their memories in order to pass on the information that they had heard. So this suggests that issues like dependence and independence become very complicated. The notion that we have a fixed synoptic tradition at this time just cannot be supported by the evidence. The sources of these gospels (if not the gospels themselves) developed within an oral environment, so what we have traditionally earmarked "redactional traces" might instead be evidence of source variation. What if the Gospel of Thomas and Luke were familiar with certain locale variants of some of the sayings of Jesus and are preserving those independently? So we have to be cautious about the redactional argument.
The big question for me is how do we distinguish between independent oral variants, and variants influenced by secondary orality (where an old independent version is adapted to the memory of another version of the saying), and secondary scribal adaptation (where an old independent version is modified by a scribe to fit his memory of the saying in another gospel or liturgy), and direct literary dependence (where Thomas took it directly from another written gospel). If anyone has any ideas, I'd love to hear them. I have run a pilot experiment within a controlled environment to try and determine whether there are certain markers or indicators or patterns characteristic of different modes of transmission (from oral to oral; from oral to written; from written to oral; from written to written). This semester I've put together a team of my students to collate the data and hope to be writing the results this summer. The experiment was occasioned by my observation that the parallels between Thomas and the synoptics are far far less verbatim than those we find in the triple or double tradition in the synoptics (see the appendix, "Verbal Similarities Between Thomas and the Synoptics" in The Original Gospel of Thomas in Translation, 2006). I'll report back on this as the data comes in.
So it's complicated, but I hope by moving the discussion away from an either/or literary dependence argument, we might actually make some progress on the problem.