The loan agreement lists the terms under which a loan is given. The bank is the lender, and any borrowing entity--individual, partnership, company--is the borrower. The loan agreement covers many of the same terms and conditions as a promissory note, although it is longer and more complicated.
The subsidiaries remain private. They are private subsidiaries of a public company. Think about it: if a subsidiary went public, that would mean that its shares were being offered to the public, instead of all being owned by the parent company, and so it would no longer be a subsidiary.
A parent company is a company that owns enough voting stock in another firm to control management and operation by doing and influencing or electing its board of directors; the second company being deemed as a subsidiary of the parent company.