Which authority regulates the conferring of degrees in South Africa?
The laws you want are the Higher Education Act 101 of 1997 and its amendments, and the NQF Act 67 of 2008 (NQF being the National Qualifications Framework).

They establish the Council on Higher Education and the South African Qualifications Authority, respectively, who between them bear responsibility for assessing, registering and accrediting institutions and qualifications in South Africa, and recognising foreign equivalent qualifications as appropriate.

I have not examined specific availability of courses, programmes, specialties or qualifications in the records maintained by either group.

However, Chapter 7 of the HEA, Act 101/1997 as amended forbids practise as a provider of higher education by anybody except "public higher education institution[s] and ... organ[s] of state" without being registered or conditionally registered, under that Act.

Section 66 of the same Act is of particular note. In full,


  66.    Offences
  
  Any person other than a higher education institution, who, without the authority of a 
  higher education institution - 
  
  (a) offers or pretends to offer any higher education programme or part thereof; 
  
  (b) purports to confer a qualification granted by a higher education institution, or in 
  collaboration with a higher education institution; or 
  
  (c) purports to perform an act on behalf of a higher education institution, 
  
  is guilty of an offence and is liable on conviction to a sentence which may be imposed for fraud. 
  
  (2) Any person who pretends that a qualification has been awarded to him or her by a higher education institution, whereas in fact no such qualification has been so awarded, is guilty of an offence and is liable on conviction to a sentence which may be imposed for fraud. 
  
  (3) Any person who contravenes section 51 (1) (a), 54 (7) or 55 (2) is guilty of an offence and is liable on conviction to a fine or to imprisonment not exceeding five years or to both such fine and imprisonment. 
  [Sub-s. (3) substituted by s. 10 of Act 54/2000] 
  
  (4) Any private higher education institution which does not comply with section 55 (1) is guilty of an offence and is liable on conviction to a fine not exceeding R20 000.


Note that 51(1)(a) is the aforementioned ban on unregistered practise, 54(7) forbids the use of names like "university" or "professor" unless entitled to use them by correct registration, 55(2) concerns deregistered institutions returning registration certificates, and 55(1) requires the public display of (a certified copy of) such a certificate.

Penalties are either specified in the section or are set equivalent to those for fraud. Needless to say, if you believe any such laws and regulations are breached, you should contact either the CHE or the SAQA directly with your information, because these are offences held equivalent to fraud.