To successfully make a parody case in regards to trademark the parody itself must make it obvious that the defendant is not in anyway connected to the owner of the trademarked thing. The Morgan and Morgan ad doesn't do that; without the disclaimer at the beginning it wouldn't be obvious that the ad wasn't a animation created by Disney. (In the original Steamboat Willie, Mickey causes a number of accidents that harm other characters. Causing a car accident would not stand out from the list of things he does.) Importantly, a disclaimer doesn't really mean much for trademark law since this body of law is about how the trademark is used.

Why does MOB get away with parodying the LV bag (in the case you cite)? Well, for starters they didn't just rip off the look of an LV bag; they stylized their renderings of the LV bags to over-emphasize aspects of the LV bags that weren't actually present in the LV bags (but were clearly evocative of the bags). They also were juxtaposing the stylized LV bag against their own product.

The case you cited also specifically refers to other LV cases in which the defendants lost (and LV won). In one case, Hyundai briefly displayed a basketball with a logo very similar to LV's trademarked monogram. Hyundai lost, even though the ad was about the ostentatiousness of luxury goods (i.e., social commentary), because nothing about the parody itself indicated that the trademark owner wasn't involved (and indeed, part of why Hyundai lost was because there was evidence that people believed that LV was making limited-edition basketballs and wanted to buy them).

More on point, the defendant in the Debbie Does Dallas case was sued by the Dallas Cowboy Cheerleaders, and lost, even though DDD is clearly a parody. The problem is that the parody was not sufficiently specific to the trademarked content (in this case, the uniform), and also it wasn't clear from the content of the film itself that the DCC wasn't connected to the film.