Part 141 is flight schools. Part 61 is independent instructors (whether or not they work for e.g. a flying club). I looked it up (so I can teach the correct answer) and found this:
Only FAA certificated Part 141 and/or 142 schools that are approved and participating in the Student and Exchange Visa Information Service (SEVIS) program are authorized to issue an I-20 form to a prospective student upon their enrollment into a course, which the prospective student needs to request the change of status and receive the subsequent student visa.
and
Part 61 flight schools and independent flight instructors may train a foreign national if that alien is a legal permanent resident or in a work status (H-type visa) with extended stay privileges or a refugee in asylum status with appropriate DHS documentation. They also may train a foreign national on an F-1 (academic visa) provided that the student is still enrolled and attending the college or university as shown on the F-1 and the student has notified SEVIS of the additional training being received at a non-SEVIS approved school.