A police spokesperson said the man was facing charges of arson, housebreaking and theft, and would appear in court on Tuesday.
Firefighters worked for hours to extinguish the blaze.
President Cyril Ramaphosa called it a "terrible and devastating event", as he vowed parliament’s work would continue.
Footage from the scene on Sunday showed a plume of black smoke filling the sky, with huge flames coming out from the roof of the building.
Officials said the fire started on the third floor offices and quickly spread to the National Assembly (the parliament’s lower house) chamber, reports the BBC’s Nomsa Maseko in Cape Town.
The parliament is not currently in session because of the holidays, and no injuries were reported.
Brigadier Nomthandazo Mbambo said the suspect is also expected to be charged under the National Key Points Act, which protects sites of strategic importance. Further details were not immediately available.
In a statement on Sunday night, South Africa’s parliament confirmed that there had been "significant damage" to the New Assembly Wing of the building, which includes the National Assembly chamber where lawmakers sit. It also said some offices had been "severely gutted".
The fire, which began shortly after 06:00 local time (04:00 GMT), came the day after Archbishop Desmond Tutu’s state funeral at St George’s Cathedral, near parliament.
Mr Ramaphosa, who visited the scene, said news of the fire was a "terrible setback to what we were basking in yesterday", and added that Archbishop Tutu would also have been devastated.
He said the building’s sprinkler system had not functioned properly, and praised firefighters for responding to the fire in minutes.
Jean-Pierre Smith, a member of the Cape Town mayoral committee for safety and security, told reporters that the roof above the old assembly hall was "completely gone" and further damage inside the old chamber had not yet been evaluated.
"It is not possible to see whether it’s damaged. We hope it is not because it has so many historical artefacts, but you can’t gain access to it without breaking the doors down and we don’t want to do that," he said.
He also said that parliament’s fire alarm only rang when firefighters were already on site.
Fire and rescue service officials earlier said it would take hours to put the fire out completely because of the carpets and wooden floors in the building.
The damage from the fire can be seen in this image made available by the city of Cape Town:
The Houses of Parliament in Cape Town are made up of three sections, with the oldest dating back to 1884. The newer sections built in the 1920s and 1980s house the National Assembly. Meanwhile, the government is based in Pretoria.
Mr Ramaphosa on Sunday spoke of his sadness at "this destruction of the home of our democracy".
Parliament, meanwhile, said "all efforts" would be made to limit any disruption to its work.
Cape Town’s Mayor Geordin Hill-Lewis said the city’s Council Chamber would be made available as an alternative place for parliament to meet.
"We want to see parliament continue its important work, and we will do what we can to help," he said.
It was the second fire at the parliament in under a year. In March there was a fire caused by an electrical fault.
Last year, a fire ravaged part of the University of Cape Town’s library, which was home to a unique collection of African archives.