I'm Dubious...

but it's not entirely out of the question. Is it "raining oil" in Louisiana?

My first guess would have been, seeing that most of the oily water shots are on pavement, that this was accumulated droplets and leaks from cars. I have no idea how long it had been since the last rain, and there is a lot of oil in this clip, but that seems the simplest explanation. It has been pointed out that under high wind conditions, droplets of oil might be blown inland as spray. See, for example, this excellent post by Dr. Jeff Masters, dissecting various aspects of hurricanes, and examining, not necessarily how potential hurricanes as a whole would interact with the oil slick, but how the various components of those storms might interact with the spill.

In the same post, he concludes that co-evaporation and condensation of oil and water is unlikely:

In general, we do not need to worry about oil dissolving into the rain, since the oil and water don't mix. Furthermore, about 50-70% of the oil that is going to evaporate from the spill does so in the first 12 hours that the oil reaches the surface, so the volatile oil compounds that could potentially get dissolved into rain water won't be around. (...) I doubt that these traces would be detectable in rainwater except by laboratory analysis, and would not cause any harm to plants or animals.

What we're seeing in the above clip is obviously far more than would be undetectable "except by laboratory analysis," and I would not allow children or animals near it. In the end, I'm not there; I don't know the situation. That being said, it seems more probable that this was spray carried inland or that the pavement just got cleaner. I'd prefer to leave hypotheses regarding hydrocarbon evaporation-precipitation cycles restricted to Titan for the time being.

Followup, Thurs. June 24: According to The CSM, the EPA concurs; there are too many unknowns to be certain it's an impossibility, but it seems unlikely at best. Additionally, the location is 45 miles from the coast, which makes spray less of a likelyhood- especially if the phenomenon wasn't seen elsewhere.