I think going during the Pageant would be great, but crowded. We've been to Nauvoo twice - during the summer both times, but have not been to the pageant. However, because it wasn't so crowded, we could get into all of the shows that we wanted and didn't have to wait for any tours.
I would say that if they want to avoid crowds, skip the pageant and go at another time during the summer. We enjoyed staying in town so we could walk around and look at the shops and everything was close. Book Hotels Now! During the summer they have a lot of performances - make sure to go to the one at night by the river. I think it is called "Rendevous in Old Nauvoo." It was so much fun! There are shows at 3 different venues, so be sure to look into all of them.
Also, we liked doing the wagon tour of the city, as it helped us get "the lay of the land." The tours of the old businesses are neat (the brickyard, bakery, cooper shop,etc.), as are the tours of the Red Brick Store and the Mansion House.
Once in Nauvoo, you aren't very far from Carthage, and that is worth the short drive.
I LOVED going to Nauvoo - I hope your parents have a wonderful time.