Short answer: Anyone who has discretionary authority over managing or administering the plan, or control over its money, is a fiduciary to that extent, whatever their title. In practice that almost always includes the employer as plan sponsor and the people or committee who make plan decisions.
ERISA defines a fiduciary by function, not by job title. A person is a fiduciary to the extent they exercise discretionary authority or control over managing the plan, exercise any authority or control over managing or disposing of plan assets, or have discretionary authority or responsibility in administering the plan.
For a self-funded plan, that usually means the employer (the plan sponsor), any named fiduciary in the plan document, and the officers or benefits committee who choose vendors, approve plan changes or decide disputed claims. If the plan document names no plan administrator, the plan sponsor is the administrator by default.
A TPA that only processes claims under rules someone else set is often not acting as a fiduciary for that work, but a TPA given discretion to decide appeals or control plan money can be. Read the administrative services agreement to see which discretion has actually been handed over.