There are several textual clues that can be used to support inferences as to why it took so long for Laurie's parents to discover the truth, but there is no direct statement made in the story that explains it.
The opening paragraph describes Laurie as a sweet-voiced nursery school tot who has turned into a swaggering, belt wearing boy who forgets to say goodbye to his mother. One could infer that one of the reasons the parents are slow to figure out the truth is because they are having difficulty letting go of the image of their sweet young son. They close their eyes to the changes going on in Laurie that go beyond his physical appearance.
The author gives several clues throughout the story that show that Laurie's behavior is similar to Charles's. He shouts raucously on two occasions: once on his first day home from kindergarten, and once the first day Charles has to stay after school. He speaks insolently to his father, saying "Hi pop, ya old dust mop" and telling a joke "Look up, look down, look at my thumb, gee you're dumb." He also spills his baby sister's milk.
But because of the invention of Charles, the parents don't suspect Laurie, they simply think that Charles has been a bad influence on their son.
It is interesting that there is no communication between adults in this situation until the PTA meeting at the conclusion of the story. The teacher doesn't call Laurie's parents to tell of his unruly behavior in school. Laurie's parents don't talk to the teacher or other school personnel about their concerns about Charles. Laurie's mother is desperate to meet Charles's mother at the PTA meeting, but she doesn't make any efforts to try to get in touch with her prior to that meeting.
The invention of Charles appears to be the perfect ruse for Laurie to deflect the truth and consequences for his bad behavior.