Jindal’s father came to the U.S. with Permanent Resident Alien status. That word “PERMANENT” counters your theory.
From the article: “Amar Jindal’s passport is notated with the code P3-1, the visa code in 1971 for “professional or highly skilled.” And Raj Jindal received her green card as his spouse.”
Even though Jindal’s mother qualified for a student visa, she was issued a green card because her husband had one.
Bobby Jindal looks to me like just an ordinary “Citizen of the United States At Birth” child of legal immigrants.
-PJ