Very few Presbyterian & Reformed seminaries are either. That is they are not "Independent" or "under" a church. Most are run by, or overseen by a board of directors that are representative of the church that they primarily serve.

The exceptions would be denominational schools that are agencies of synod or GA. The two most prominent examples in this catagory are schools such as Covenant (PCA) or Erskine (ARP). In these cases the schools are still run by boards, however the broader court has a right of review of past actions & have an advisory position re the future on an annual basis.

Others such as Greenville, WTS, Know, etc. are all board run schools. I know of no Reformed seminaries that are under the authority of a presbytry or a session.

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Kevin Rogers
Sovereign Community Church, PCA
Moncton NB