Some do special offers - I believe that the Disney parks do something, and Knott's Berry Farm do special buffet dinners, but assuming this isn't happening, are the crowds light that day?
(I understand that the weekend of Thanksgiving is likely to be busy.)
They will be very, very busy. So get in early. Epcot in Disneyland can handle large crowds.
I'm going to answer my own question (partially) here, because I went to Six Flags Magic Mountain on Thanksgiving Day this year.
It was moderately busy, almost entirely with young adults. There were very few children there.
The lines on the more extreme/newer rides were long - 45 minutes or so. The lines on the more moderate rides (e.g. Revolution, an older 'coaster with a single loop) which were not so popular amongst adults were almost non-existent.
Thanksgiving Week is one of the most crowded weeks of the year at Walt Disney World. A good rule of thumb is that Disney World will always be more crowded than usual when students have time off of school. So summer is more crowded than the school year. And holidays are more crowded than the summer. The more common the holiday, the more crowded the park will be (ie: Thanksgiving Week will be more crowded than Presidents Day weekend, but both will be more crowded than a generic summer day).
