The two adults were spotted four times at Warley Woods after the noise of the engine woke up nearby residents, who then reported them to the police.
The incidents happened between January 18 and 28, and the antics of those responsible were captured on CCTV cameras in the area.
Warley Woods Community Trust manager Viv Cole said it was the latest example of anti-social behaviour at the parkland off Lightwoods Hill and Abbey Road in Bearwood.
"Just last week somebody was riding motorbikes over the greens destroying it," she said.
"We have been working out how to tackle it and we're hoping they will simply ruin out of petrol eventually and stop doing it.
"They were coming a couple of nights a week. We know they come in the early hours and from the cameras in the area we know it's not teenagers doing this.
"Residents neighbouring the park have been woken up by the noise and they have been calling the police to report it.
"It's very important for people in the area to continue to report incursions as the more reports are made, the more information we can get to try to identify who's responsible.
"It's two people who've been riding round in circles across the grass.
"This time of the year the grass doesn't grow and we don't even let the players take golf buggies to the greens in an effort to protect them."
West Midlands Police confirmed that it was aware of the latest vandalism.
In a statement the force said: "We're aware of some recent incidents of anti-social behaviour and criminal damage at Warley Woods Golf Course, in particular people riding off-road bikes and quad bikes on the course in the early hours of the morning.
"This behaviour is illegal and it is not acceptable.
"The local neighbourhood team have contacted Warley Woods Golf Club and officers will be patrolling the area.
"We’d absolutely encourage anyone with information about these incident and the people involved to get in touch with us."
In 2016 the trust was forced to close the gates at night again due to worries over an influx of travellers with caravans. The previous year £3,000 was raised to replace picnic benches which were damaged by vandals.
In 2014 groups driving in after dark and causing a nuisance by playing loud music also prompted gate closure at the site which is listed in the Keep Britain Tidy top 10 most popular places to visit.
Anyone with information should phone the police via 101 quoting crime number 20/176290/22.