Trademark dilution is a term you may not have heard before. It refers to using a trademark in a way that "dilutes" the owner's trademark. Dilution doesn't just happen when someone infringes on a trademark directly. Parodies of trademarks could result in dilution as well.
In 1996, the Trademark Anti-Dilution Act began to allow owners of trademarks to seek injunctive relief to prevent their trademarks from being diluted. The act, by law, makes sure famous marks aren't diluted whether there is competition or the likelihood of confusion.
One example is from a past Wendy's International, Inc. v. Big Bite, Inc. case. In this case, Big Bite had used a red-haired girl, among other satirized trademarks, in advertisements. The court recognized Wendy's complaint that Big Bite hadn't considered the potential for consumer confusion and stopped the further use of the ads.
If a person's parody is strong enough to obviously not dilute the owner's trademark, then a court will allow it. On the whole, if businesses can't show that there is a likelihood of confusion, the courts are more likely to rule in favor of the person who created the parody. If the people creating parodies do not make the parody significant enough to eliminate the likelihood of confusion, then the court can rule that there was trademark infringement.
It can be difficult to know where you stand in trademark law, since parodies are allowed in many different cases with relative frequency. Cases like this become complicated, which is why it's wise to look into all your legal avenues.