Dlamini was injured when Table Mountain National Parks rangers detained him for allegedly failing to pay the entrance fee at the Silvermine section of the park where he was cycling on Friday.
SANParks said it was in the process of appointing an independent service provider to conduct a preliminary investigation into the incident.
Environment, Forestry, and Fisheries Minister Barbara Creecy took to Twitter on Friday, saying she instructed that the rangers involved in the altercation be suspended.
The NTT Pro Cycling Team said Dlamini was transferred for further specialist consultation together with the team’s medical department.
SANParks’ acting communications head, Rey Thakhuli, said the organisation was taking the matter seriously.
“We wish him a speedy recovery and we will do everything in our power as SANParks to assist him and his family during this time,” he said.
NTT Pro Cycling said SANParks needed to engage with Cape Town’s cycling community to establish a positive relationship going forward. They also wanted the organisation to issue a public apology to both Dlamini and the eyewitness who recorded the incident.