At the end of Episode 1, Tenchi's family isn't the least bit against him going into space on a career that may kill him, and they even make a melodramatic farewell video for him to watch when he wakes up in space! ....when they had no idea what the application was for, and made him sign it anyway! Heck, her sister even gets his room!
The fact that Seina was more upset that his family was eating the cake Kimiko made him than them selling him out.
Crowning Moment of Funny (After being admitted into the GP academy, Seina is chosen as a representative of the freshman class. When he gets up before the whole school to make a speech, his luck causes the academy speaker system to dial itself up to eleven, resulting in the entire place being leveled when Seina opens his mouth.)