Ms. Heath is among those who help care for Trig, 2, who has Down syndrome. She also helps go through the hundreds of letters Ms. Palin receives asking her to speak at or attend various events.

Mr. Heath said he was careful about nudging his daughter toward the things that interest him.

“I don’t want to push the wrong button with Sarah,” he said. “Besides, she doesn’t make the decisions. Let me retract that. I’m sure she thinks them over and she has a lot of say as to yes and no.”