The Comfort Diner, with two locations in Manhattan, is a "theme restaurant" that works. The decor is deliberately 50's and the menu contains dishes like "Soup and Sammy" or "a Nice Healthy Salad," but thanks to the lighthearted spirit and the classic food they manage to pull off the theme perfectly. There are daily blue plate specials, good basic meals like yankee pot roast and turkey with trimmings, decent desserts, and fun beverages like egg creams and cherry lime rickeys. The milkshakes are also reputed to be fabulous, and often come in imaginative special flavors. When my parents ask to "eat at that one place" during each visit, this is the place they mean.
One warning: lines are long during the lunch rush and at weekend brunch, and both locations can get noisy when they're crowded. The midtown location is a few blocks from public transportation (Grand Central Station has the closest subways), but the Upper East Side location has a 4/5/6 line at its door.