Is it permissible to skip two academic years by directly enrolling in the third year of secondary school, given the ability to review the missed studies?

You should refer to the educational institution's regulations: If the regulations permit what your friend mentioned, then there is no harm in it.

If there are no regulations, or if they exist but do not explicitly permit or prohibit it: It is permissible to accept your friend's suggestion, provided that the director is authorized to approve it, and that their approval is free from favoritism or bribery.

If the above is not possible, and adopting this course of action would involve a violation of Sharia regulations: Then it is not permissible to do it, and one must fear Allah and be patient.