It's unclear whether Fox actually intends to bring legal action over what appears to be a fairly obvious case of copyright infringement, but if history is any indication they're legal department is probably seriously considering it. In 2003 Fox News reportedly considered suing Simpsons creator Matt Groening over a parody of the news channel on sister service Fox Broadcasting. The suit was never filed due to Groening's long and profitable relationship with Fox.
Although parodies are generally covered by fair use provisions, since this press release parodies the Democratic supporters of the bill being discussed, rather than being a parody of the copyrighted characters.
Sources:
Republican Press Release
Techdirt