This tale of his Constitutional punishments is what might be called an "apocryphal tale" and, like other apocryphal tales, a search for evidence is the only way to prove or deny the tale. What is true and established in Gibson's biography of Marshall's first 30 years of life is that he and his father had a unique relationship, firstly, because William Marshall was a uniquely intelligent and talented man and, secondly, because Thurgood was an equally unique, intelligent and talented youth.
Kirkus literary reviewers summarize Gibson's description of Thurgood's relationship with his father by explaining that they unhesitatingly shared conversations about the issues of the day in politics and race. Both having lived under segregation laws, these conversations molded Thurgood's future visions.
Growing up, Marshall intensely discussed politics and race relations with his father. “He never told me to be a lawyer, but he turned me into one,” Marshall later said. (Kirkus Reviews)
As to the story of reading the U.S. Constitution during punishment times at school, this is a story that Thurgood seems to have offered as a means of explaining his early mastery on the Constitution through self-effacing humor. Quite possibly, he wanted to keep the truth to himself out of modesty or some other equally benign motive, though my suggestion here is pure speculation.
The apocryphal tale about Marshall's slave ancestor being set free by his master because he was too uncooperative to be a slave grew in detail over the years but does not withstand scrutiny. Likewise, Marshall's story about his learning the US Constitution when he was sent to the basement of his elementary school or high school for punishment is doubtful. (Gibson, Young Thurgood)