Furore, sarcasm, a cartoon, the former governor of Alaska, and Down syndrome

When people with disabilities play characters in a sitcoms that lampoon politicians, celebrities, and everyone else on earth, do they automatically lampoon themselves as people with disabilities?
In a recent episode of Family Guy, an animated American sitcom known for unforgiving sarcasm and satire, a girl named Ellen, who has Down syndrome, says that her mother is “the former governor of Alaska.”
From the New York Times:

Ms. Palin, whose son Trig also has Down syndrome, has said that the “Family Guy” show “really isn’t funny” and was the work of “cruel, cold-hearted people.” Ms. Palin’s daughter Bristol has written that the “Family Guy” writers were “mocking my brother and my family,” and called them “heartless jerks.”

The actor who provided the voice for Ellen, 39-year Andrea Fay Friedman, has accused Ms Palin of having no sense of humor. Ms Friedman has Down syndrome.
Here’s what she said in an interview with NYT:
Q. When you get asked to play characters who have Down syndrome, does that make you at all uncomfortable?
A. No, I’m proud of it. I’m not embarrassed. But mostly, it doesn’t matter if you have Down syndrome. Really, it just matters to have a different challenge.
Q. When did you find out about the reaction that the episode elicited from Sarah Palin and her family?
A. [laughs] That I did not even know about until my mom told me, “You’re on Channel 4!” And when I watched on Channel 4, on “Extra,” and I saw Sarah Palin with her son Trig. I’m like, “I’m not Trig. This is my life.” I was making fun of Sarah Palin, but not her son.