Tonya Couch and her son Ethan were taken into custody this week in Mexico.
Known as an "amparo," or protection, such appeals try to block a government action. Couch's attorneys are asking a judge to prevent authorities from deporting him or holding him without contact with lawyers, family members or visitors.
18-year-old Ethan Couch won a three-day court injunction.
Ethan Couch and his mother, Tonya Couch stayed first at a resort known as Los Tules, near the beach.
During the sentencing phase of his trial, a defense expert argued that his wealthy parents coddled him into a sense of irresponsibility - a condition the expert termed "affluenza."