Fourteen adult birds and a number of eggs were removed from Millwall Park last month, prompting Islanders to launch a campaign to stop the cull, claiming they are feathered friends not parrot pests.
Two petitions have been started against the Department for Environment, Food and Rural Affairs' (Defra) policy, while residents are threatening direct action to stop culling teams accessing remaining nests.
She said: "It's terrible what Defra are doing. These birds have been here for years and the locals love seeing them here. They are part of the Island's wildlife and very friendly, so we're shocked at what's happening.
"Three lorries turned up without warning at the Park and used a cherry picker to remove eggs from the nests. They caught some of the adult birds in nets and then smashed up the nests so others couldn't return to them.
"For days afterwards the birds were screeching and flying around looking for their eggs and their nests. It was really upsetting.
"The children at George Green's School are also upset about it and have started their own petition to send to Defra. Some of the kids have even threatened to tie themselves to the trees to stop the nests being removed.
"We'll block the roads to stop the lorries coming back to get the rest of the birds if we have to."
The humane cull, which sees birds caught and rehoused in aviaries rather than being killed, was announced by Defra earlier this year.
Although there are only about 150 monk parakeets in the UK, Defra decided on the cull because their large nests - often built on telegraph poles and phone masts - can cause damage.
Mr Smith said: "We'd understand it if these nests were on telegraph poles, but they are in the trees, hidden away and not causing any damage.
"These birds have been here for about 10 years and they haven't multiplied out of control, there can't be more than two dozen in the whole area.
"We've already got hundreds of people supporting the petition. All species in this country have evolved somewhere else and been accepted here. The parakeets are no different."
A Defra spokesman said: "Control work is being carried out to counter the potential threat monk parakeets pose to critical national infrastructure, crops and native British wildlife.
"This invasive species has caused significant damage in other countries through nesting and feeding activity and we are taking action now to prevent this happening in the UK.
"Trained staff will use the most humane method appropriate in individual situations. This will involve a range of techniques such as trapping, shooting and nest removal. Birds will be captured alive and re-homed whenever possible."
The cull is backed by the Royal Society for the Protection of Birds.
Spokesman Tim Webb said: "We are a conservation agency and have campaigned against lethal control measures. The Government agreed with us and that's why the capture and rehousing policy is being followed.
"Some control is required. These colonies come from birds kept as pets that are released into the wild. We'd like to see the law changed to stop the import of birds into this country, that would be the best control."
The monk parakeet is native to South America, where it has been known to cause damage to crops because of its numbers.
It has bright green feathers on its upper parts, and distinctive blue wing feathers.
The monk parakeet builds a stick nest, which it adds to, causing the structures to grow excessively large. Some nests have been as big as small cars.
The monk parakeet is also known the Quaker parrot. It is smaller than its cousin, the ring-necked parakeet, which is also found in southern Britain.
Around 150 monk parakeets live in the home counties and London.