Culture Minister Carál Ní Chuilín was taken to hospital following the incident with a police land rover in the Carrick Hill area in the north of the city.
Sinn Féin's Gerry Kelly was also at the scene which followed the Orange Order's annual Tour of the North.
The parade, which began at 7pm, passed through the mainly nationalist area, with around 1,000 marchers and 15 bands taking part.
Police said the march was "largely peaceful" but added that a small number of nationalist youths tried to run through the bands as they made their way up Peter's Hill.
Following a 15 minute stand-off, police sealed off part of the area in an attempt to keep rival groups apart.
I can't think of any circumstances where it's acceptable police behaviour to drive off with somebody on the bonnet of a jeep.
Gerry Kelly, Sinn Féin
Mr Kelly, MLA for north Belfast, said he was talking to young people who had agreed to go home when he was made aware that a teenager had been arrested and was inside a police land rover.
The politician said he spoke to the officer driving the vehicle, who indicated he would pull over so they could speak with the teen's parents.
Instead, Mr Kelly told UTV, the vehicle drove off - an action which he said "annoyed the crowd and I have to say it annoyed me".
He said: "I was there as an elected representative, a member of the Policing Board, and I was told lies. I don't like that."
"I then stood in front of the fourth jeep and demanded to see whoever was in charge. At that stage he rolled the jeep into me.
"I held on to the grill to get my balance because I near fell over. When I held on to the grill he drove off.
"It felt like a lot longer but I suppose he drove maybe 20 yards down the road," Mr Kelly added.
In the minutes afterwards, Mr Kelly said his party colleague, Ms Ní Chuilín, was injured as she came to his aid.
He explained: "Carál had run after the jeep and got injured as a matter of the jeep hitting her as well. She's got torn ligaments in her shoulder."
A PSNI spokesperson has revealed the incident will be investigated by the Police Ombudsman.
However, the Sinn Féin representatives say they remain angry at what happened.
Ms Ní Chuilín said: "I got injured when the jeep was driving off and we are very, very lucky there aren't more serious injuries if not worse in this community.... I was hit by the jeep.
"There was a bit of recklessness here last night by some of the police in those jeeps. That's not conduct befitting of PSNI and we will certainly be challenging it."
Standing in front of a police land rover that is removing someone who has been arrested - that is not what should be happening.
Nelson McCausland, DUP
Mr Kelly said the incident has damaged local residents' confidence in policing.
"I will be taking this further," he added.
"What I'm very annoyed at is the driver of this jeep, and whoever was in charge of him doing something which he should not have done.
Mr Kelly commented: "It's bad policing. It does knock things back in this area."
Nelson McCausland, DUP MLA for the area, said it had been a successful Tour of the North, however he was disappointed to hear of the disturbance involving the police and Sinn Féin representatives.
"It was particularly disappointing when one of them is a member of the Northern Ireland Policing Board," he said of Mr Kelly.
"But obviously this will now have to be investigated fully by the police."
Mr McCausland added: "I think it's inappropriate for politicians to interfere with policing in that sort of way.
"We should respect law and order and that should simply not be on the agenda for any politician."
Two males, aged 20 and 16, were arrested during disturbances in the nearby Peters Hill area following the parade.
It is also alleged band members breached Parades Commission determinations and a senior police chief said they are investigating "a number of incidents".