On the best-seller wall of my local bookstore is Mo Willems’s new picture book, “The Pigeon HAS to Go to School!” My children adore Mr. Willems’s books, but he’s got one thing wrong: The pigeon doesn’t have to go to school.

While Democratic presidential candidates tout subsidized early schooling and cities like New York expand universal preschool to 3-year-olds, some parents are resisting the pressure to start school early. For Dawn Gallahue, enrolling her 4½-year-old son, Noah, in preschool didn’t seem like a good fit, and...