How is the system supposed to work?
In their last year of high school, students are supposed to make a number of "wishes", wishes being for a particular course at a particular school. Based on their grades and other factors, schools select the students for their courses. Students then have to make a final choice of where to go based on which of their wishes were accepted.
Without getting into the finer detail of French higher education, the short of it is that the standard university courses are largely non-selective, and everything else is selective to very selective. One such type of selective courses is "preparatory classes", or prépas, courses that prepare entry into postgraduate schools. These prépas are offered by high schools, so it is possible to stay at your high school after high school. This is what the minister's son did: he entered a prépa at his high school, Stanislas.
Obviously, making a single wish for a selective course is extremely risky, because you might end up not being selected there, and therefore anywhere. But simply doing that is only a problem for yourself.
How is Stanislas breaking the system?
Stanislas is accused of offering a guarantee of admission to some students in exchange of them making a single wish. Students are guaranteed a school, and the school is guaranteed a (paying) student.
This first means the admissions are rigged. The number of spots actually opened to other applicants is less than the number of spots nominally offered, since the school is already filling those spots before the admissions process is even open. Equality of chances is supposed to be a founding principle of our school system, this scheme makes a mockery of that. Also, Stanislas received public funding, so they can't just flout the rules as they see fit.
This also means there is a significant opportunity for corruption. This scheme isn't offered to every student. It's an opaque scheme, offered to some students only, outside of the normal process. The value of this for a student/parent can be quite high, as a good prépa opens a lot of good doors. At the very least this creates a conflict of interest if one such student/parent was to have any sort of business with the school.
And it looks particularly bad for Stanislas because, from a total of 600,000 highschoolers, only 41 students in France made only one wish for a prépa at their current high school in 2023, and 38 of them were Stanislas students, including the minister's son (source Mediapart, can't find an accurate non-paywalled quote for it). It is an interesting statistical anomaly for sure.
So why is it bad?
Why is it a bypass? Because it just is. Schools aren't supposed to pick their students before the selection process even starts. It is bad because Stanislas receives public funding and that means they have to follow the rules.
Why is it corruption? Kinda depends for whom and what specific crime we're talking about.
Consider the facts:
Stanislas has been under investigation by the ministry of education (for accusations of institutionalised homophobia, disinformation on abortion, proselytism, moral or even sexual abuse, and other things). The investigation started in February 2023, the report was delivered to the ministry in the summer of 2023. No decision has been made yet.
Amélie Oudéa-Castéra became minister of education this January 2024. She was (and still is, don't ask) minister of sports since May 2022. She is the third minister of education since the summer of 2023.
Her son was admitted for a prépa at Stanislas last year, in spring of 2023. He made only one wish, and the minister has publicly recognised that the school offered her son a spot if he made just this one wish.
This means there is a conflict of interest. The minister has a personal stake with the school, with her son being admitted at the school. That the minister's son benefited from a less-than-legal scheme to ensure his admission further aggravates the conflict of interest.
That the minister's son benefited from that scheme definitely implicates the school. This might be considered some kind of insider trading, or analoguous. It may not implicate the minister or her son, as she claims to "have followed the procedure given by Stanislas and every step of ParcoursSup", "like other families" (sources Mediapart, quoted here).
Regarding the conflict of interest, the minister already recused herself from dealing with the Stanislas report or exerting any influence on its resolution. I don't think anything can make it look less bad politically, but that much isn't a crime, nor is the simple existence of a conflict of interest.
For it to become a case of good old fashioned corruption, several things would have to be established. That the school intended to have the minister in their pocket, e.g. to have a minister of the government speak favorably on their behalf while under investigation by a government agency. And then for the minister to be implicated, she would have had to know and agree to such quid pro quo. So far, no element indicates any such thing happened.