1. Contracting Parties: All parties to a contract are contracting parties. Likewise all parties to a contract are partes contratantes/contratantes. Vocabulário Jurídico, De Plácido e Silva, 26ª Ed., pág. 373. However, the Dicionário Marcílio refers to a second usage of contratante, i.e. when used in contradistinction to contratada. There is no equivalent usage in English.
2. Contractor: The standard usage is that a contractor is a person who contracts to do work for another person, but strictly speaking a contractor is any person who enters into a contract, i.e. the same as contracting party. Blacks Law Dictionary, 6th Ed., p. 326.
3. Contracted party is equivalent to Contratada and means e.g. "the consultants, contractors, and their subconsultants and subcontractors and any other party providing material or services", i.e. a broader meaning than the standard meaning of contractor.
https://www.lawinsider.com/dictionary/contracted-party4. Accordingly, Contracting Party is the "odd man out" and is not equivalent to Contratante.