Heavy gunfire could be heard on an audio recording from inside the parliament chamber, and there are reports that a security guard has died.
Several people were injured at the shrine, dedicated to the founder of the modern Islamic republic.
There was one suicide bomber and two or three other gunmen.
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Ali Khalili from the mausoleum's public relations office told IRNA news agency that one of the armed men detonated himself in front of a bank outside the mausoleum.
However Fars News said that the suicide bomber was a woman.
The Iranian Labour News Agency reported that two attackers had been arrested.
Five people were wounded at the shrine, and one person confirmed dead, it said.
Latest reports suggest the attack at Iran's parliament is ongoing.
An image posted by Tasmin News showed what appeared to be a gunman holding an assault weapon shooting out from an upstairs window of the parliament building.
MP Seyyed Hossein Naqavi-Hosseini told the Iranian Students' News Agency (ISNA) that three gunmen were still in the building, in MPs' offices.
Iranian rolling news channel IRINN said seven or eight people had been injured at the parliament by four attackers.
The gunmen were carrying Kalashnikovs and one Colt pistol, reports said.
One gunman has been arrested, IRNA said.
The attack at the mausoleum, several kilometres to the south, happened at the same time and appeared to be co-ordinated, reports said.
Armed individuals have opened fire at the Iranian parliament and the shrine of Ayatollah Khomeini in the capital, Tehran, causing a number of casualties.